From Heun Class Equations to Painlevé Equations
In the first part of our paper, we discuss linear 2nd order differential equations in the complex domain, especially Heun class equations, that is, the Heun equation and its confluent cases. The second part of our paper is devoted to Painlevé I-VI equations. Our philosophy is to treat these families...
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| description | In the first part of our paper, we discuss linear 2nd order differential equations in the complex domain, especially Heun class equations, that is, the Heun equation and its confluent cases. The second part of our paper is devoted to Painlevé I-VI equations. Our philosophy is to treat these families of equations in a unified way. This philosophy works especially well for Heun's class equations. We discuss its classification into 5 supertypes, subdivided into 10 types (not counting trivial cases). We also introduce in a unified way deformed Heun class equations, which contain an additional non-logarithmic singularity. We show that there is a direct relationship between deformed Heun class equations and all Painlevé equations. In particular, Painlevé equations can also be divided into 5 supertypes and subdivided into 10 types. This relationship is not so easy to describe in a completely unified way, because the choice of the ''time variable'' may depend on the type. We describe unified treatments for several possible ''time variables''.
|
| first_indexed | 2026-03-15T05:48:35Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 056, 59 pages
From Heun Class Equations to Painlevé Equations
Jan DEREZIŃSKI a, Artur ISHKHANYAN bc and Adam LATOSIŃSKI a
a) Department of Mathematical Methods in Physics, Faculty of Physics,
University of Warsaw, Pasteura 5, 02-093, Warszawa, Poland
E-mail: jan.derezinski@fuw.edu.pl, adam.latosinski@fuw.edu.pl
b) Russian-Armenian University, 0051 Yerevan, Armenia
E-mail: aishkhanyan@gmail.com
c) Institute for Physical Research of NAS of Armenia, 0203 Ashtarak, Armenia
Received August 25, 2020, in final form May 25, 2021; Published online June 07, 2021
https://doi.org/10.3842/SIGMA.2021.056
Abstract. In the first part of our paper we discuss linear 2nd order differential equations
in the complex domain, especially Heun class equations, that is, the Heun equation and
its confluent cases. The second part of our paper is devoted to Painlevé I–VI equations.
Our philosophy is to treat these families of equations in a unified way. This philosophy
works especially well for Heun class equations. We discuss its classification into 5 supertypes,
subdivided into 10 types (not counting trivial cases). We also introduce in a unified way
deformed Heun class equations, which contain an additional nonlogarithmic singularity.
We show that there is a direct relationship between deformed Heun class equations and all
Painlevé equations. In particular, Painlevé equations can be also divided into 5 supertypes,
and subdivided into 10 types. This relationship is not so easy to describe in a completely
unified way, because the choice of the “time variable” may depend on the type. We describe
unified treatments for several possible “time variables”.
Key words: linear ordinary differential equation; Heun class equations; isomonodromy defor-
mations; Painlevé equations
2020 Mathematics Subject Classification: 34A30; 34B30; 34M55; 34M56
Contents
1 Introduction 2
2 Second order linear differential equations in the complex domain 6
2.1 Differential equation and operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Singularities of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Singularities of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Sandwiching with functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Half-integer rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 Simplifying the equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.7 Solutions in terms of formal power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.8 Thomé solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.9 Nonlogarithmic singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Equations with rational coefficients 17
3.1 The Mn class and the grounded Mn class . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Generalized Fuchs relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Riemann class equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Heun class equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5 Deformed Heun class equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.6 Classification of Heun class equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
mailto:jan.derezinski@fuw.edu.pl
mailto:adam.latosinski@fuw.edu.pl
mailto:aishkhanyan@gmail.com
https://doi.org/10.3842/SIGMA.2021.056
2 J. Dereziński, A. Ishkhanyan and A. Latosiński
4 From Heun class to Painlevé equations 24
4.1 Method of isomonodromic deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 Isomonodromic deformations in presence of a non-logarithmic singularity . . . . . . . . . 25
4.3 Isomonodromic deformations of Heun class equations . . . . . . . . . . . . . . . . . . . . . 26
4.4 Correspondence between Heun class and Painlevé equations . . . . . . . . . . . . . . . . . 30
4.5 From Heun (111;1) to Painlevé VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.6 From Heun (21;1) to Painlevé V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.7 From Heun (3
21;1) to degenerate Painlevé V . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.8 From Heun (2;2) to non-degenerate Painlevé III′ . . . . . . . . . . . . . . . . . . . . . . . 34
4.9 From Heun
(
2; 3
2
)
to degenerate Painlevé III′ . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.10 From Heun
(
3
2 ; 2
)
to degenerate Painlevé III′ . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.11 From Heun
(
3
2 ; 3
2
)
to doubly degenerate Painlevé III′ . . . . . . . . . . . . . . . . . . . . . 36
4.12 From Heun (1;3) to Painlevé IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.13 From Heun (1; 52 ) to Painlevé 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.14 From Heun (;4) to Painlevé II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.15 From Heun (;72 ) to Painlevé I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5 Five supertypes of Painlevé equation 39
5.1 Overview of five supertypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Painlevé V or (112) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.3 Painlevé III′ or (22) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 Painlevé IV-34 or (13) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.5 Painlevé I–II or (4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
A Proof of Theorems 4.1, 4.2 and 4.3 47
A.1 Preparation for the proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
A.2 First compatibility condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
A.3 Second compatibility condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
A.4 Case A, Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
A.5 Case A, Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
A.6 Case B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
B Hamilton equations 56
B.1 From Hamilton equations to second order equations . . . . . . . . . . . . . . . . . . . . . 56
B.2 Invariance of Hamilton equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
B.3 Hamiltonian solvable in quadratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
References 58
1 Introduction
There are many types of differential equations and special functions. Typically, within a given
class there is one generic type and many confluent types. This is the case of Riemann (hyper-
geometric) class equations, Heun class equations, as well as Painlevé equations. For instance,
the generic type of Riemann class equations can always be reduced to the Gauss hypergeomet-
ric equation, but there are also confluent types such as Kummer’s confluent equation, the F1
equation, and the Hermite equation, see, e.g., [2, 3, 17, 24, 25].
One can try to understand the process of confluence by considering equations depending
holomorphically on parameters. Some properties of the whole class can be described in a uniform
way, without splitting the class into types. For instance, one can identify various transformations
(“symmetries”) that leave the class invariant.
In order to study the equations in more detail, one needs to split the class into types. Within
a given type one can simplify the equation by symmetries and convert it to a normal form,
thereby reducing the number of parameters. This has to be done case by case.
From Heun Class Equations to Painlevé Equations 3
In the case of hypergeometric class equations, this idea was successfully applied in the book
by Nikiforov–Uvarov [18]. It works especially well for hypergeometric polynomials, that is Jacobi,
Laguerre, Bessel and Hermite polynomials, which can be elegantly treated in a unified way.
In this paper, we try to apply this idea to the derivation of Painlevé equations from Heun class
equations by the method of isomonodromic deformation. We will see that each type of Heun
class equation corresponds to a (properly understood) type of Painlevé equation. The passage
from Heun class to Painlevé can be accomplished in a fairly uniform way, although one has
to consider several similar but distinct cases.
We start our paper by Section 2 containing basic theory of singularities of 2nd order scalar
ordinary differential equations. This is a classic subject with several well-known textbooks, such
as [9, 23]. We follow to a large extent the treatment described in the monograph by Slavyanov–
Lay [25] and the appendix to [24], written by Slavyanov. Sometimes we introduce new notation
and terminology to make precise some concepts which in [25] are implicit.
The central concept of the theory of linear differential equations in the complex domain is
the rank of a singularity. In our paper we introduce several kinds of the rank. In particular,
we distinguish between the usual rank and the absolute rank (the infimum over the ranks of all
possible transformed forms of a given equation). The rank can be an integer or a half-integer.
We also introduce the rounded rank, which has always an integer value. Thus if the rank is m
or m − 1
2 , where m is an integer, then we say that its rounded rank is m. In particular, the
rounded rank is 1 if the singularity is Fuchsian (also called regular). We believe that all these
concepts clarify the theory of ordinary differential equations. We also discuss formal power series
solutions of these equations (the so-called Thomé solutions). We introduce the concept of indices
of a singular point. This is of course well-known for Fuchsian singularities. For non-Fuchsian
singularities this concept is not so well known, although it is implicit in [25].
In Section 3 we discuss equations with rational coefficients. Such equations have a finite
number of singularities on the Riemann sphere. Following [25], the class of equations with n
singularities in C and a singularity at infinity, and their confluent cases are called Mn class
equations. red It is easy to see that these equations can be written as(
σ(z)∂2z + τ(z)∂z + η(z)
)
u(z) = 0, (1.1)
where σ is a polynomial of degree ≤ n, τ of degree ≤ n − 1 and ση of degree ≤ 2n − 2.
We introduce also a closely related grounded Mn class, for which one of indices of all finite
singularities is zero — hence the name “grounded”. (In [25] such equations are called canonical.
In our opinion, the word canonical is overused, hence not appropriate for this meaning.) They
can be written as (1.1) with the same conditions on σ, τ , and with η a polynomial of degree
≤ n− 2.
The best known classes of equations with rational coefficients are the M2 class and the
groundedM2 class. We callM2 the Riemann class, since its generic representative is the Riemann
equation with one singularity at∞. The grounded M2 class is especially often encountered in the
literature. It is the main subject of the textbook by Nikiforov–Uvarov [18], where its elements are
called “hypergeometric type equations”. Note that the difference between the full and grounded
Riemann class is minor — the only type of equations contained in the full Riemann class but
not represented in the grounded Riemann class is the Airy equation.
One of the central objects of our paper is the M3 class, called also the Heun class, for which σ
is a polynomial of degree ≤ 3, τ of degree ≤ 2 and ση of degree ≤ 4. The main type within the
Heun class is the standard Heun type, that is the equation with 4 Fuchsian singularities in the
Riemann sphere, one of which put at ∞. It was first analyzed by Heun [8], see also [16] for
a recent study.
In the literature the name Heun class equations is employed in two meanings: the meaning
that we have just described is used in [24, 25]. It is also common to use it for what we call the
4 J. Dereziński, A. Ishkhanyan and A. Latosiński
grounded Heun class, see, e.g., [5]. The grounded Heun class (not counting types of the Riemann
class) is divided into 5 types: standard, confluent, biconfluent, doubly confluent and triconfluent.
The full Heun class, beside the above five types has five more types, which we call degenerate con-
fluent, degenerate biconfluent, degenerate doubly confluent, doubly degenerate doubly confluent
and degenerate triconfluent. ([25] uses the word “reduced” instead of “degenerate”.)
Sometimes a singularity of an equation does not lead to a singularity of its solutions. We call
such points non-logarithmic singularities. Non-logarithmic singularities play an important role
in the following construction.
Let us start with an equation (1.1). We introduce the deformed form of (1.1) to be the
equation(
σ(z)∂2z +
(
τ(z)− σ(z)
z − λ
)
∂z + η(z)− η(λ)− µ2σ(λ)− µ
(
τ(λ)− σ′(λ)
)
+
µσ(λ)
z − λ
)
× v(z) = 0. (1.2)
Note that all finite singularities of (1.2) are the same as in (1.1), except that there is an additional
non-logarithmic singularity with indices 0, 2. λ is the position of this singularity and µ = v′
v (λ).
If (1.1) is a Heun class equation, then (1.2) is called a deformed Heun class equation [25].
Let us note that non-logarithmic singularities of an equation can be produced by a change
of gauge, that is by replacing the unknown v(z) with c0(z)v(z) + c1(z)v
′(z), where c0, c1 are
some rational functions. This procedure applied to grounded Heun class equations produces the
so-called derivative Heun class equations, see [5, 10, 26]. Kimura proved in [15] that in deformed
Riemann class equations one can remove all non-logarithmic singularities by a change of gauge.
We do not know if the same is possible for all kinds of deformed Heun class equations.
Painlevé equations is a famous class of nonlinear differential equations with the so called
Painlevé property–the absence of moving essential and branch singularities in its solutions. They
were discovered in the beginning of 20th century [7, 21, 22]. Traditionally, Painlevé equations
are divided into 6 types, called Painlevé I, II, III, IV, V and VI [12]. As noted by Ohyama–
Okumura [19], it is actually natural to subdivide some of them into smaller types, obtaining
altogether 10 types. Each of them corresponds to one of types of the Heun class equations.
We obtain the following correspondence between types of the Heun class and Painlevé:
(1111) (standard) Heun Painlevé VI
(112) confluent Heun Painlevé nondegenerate V(
11 3
2
)
degenerate confluent Heun Painlevé degenerate V
(22) doubly confluent Heun Painlevé nondegenerate III′(
3
22
)
degenerate doubly confluent Heun Painlevé degenerate III′(
3
2
3
2
)
doubly degenerate doubly confluent Heun Painlevé doubly degenerate III′
(13) bi-confluent Heun Painlevé IV(
1 5
2
)
degenerate bi-confluent Heun Painlevé 34
(4) tri-confluent Heun Painlevé II(
7
2
)
degenerate tri-confluent Heun Painlevé I
Above, the symbols such as (112) indicate the ranks of the singularities in the Heun class
equation. We use underline to indicate the rounded rank: thus
1 denotes either 1 or
1
2
(a Fuchsian singularity); 2 denotes either 2 or
3
2
;
3 denotes either 3 or
5
2
; 4 denotes either 4 or
7
2
.
From Heun Class Equations to Painlevé Equations 5
Following the literature, instead of Painlevé III we prefer to use an equivalent equation
Painlevé III′.
Note that the Painlevé deg-V is equivalent to Painlevé III′ and Painlevé 34 is equivalent to
Painlevé II by a relatively complicated change of variables.
As noted by Ohyama–Okumura [19], there also exists another, coarser classification of Pain-
levé equations into five supertypes. It corresponds to a coarser classification of Heun class
equations into supertypes, where we use rounded ranks instead of the usual ranks. We obtain
the following table, which is discussed in detail in Section 5:
(1111) (standard) Heun Painlevé VI
(112) confluent Heun Painlevé V
(22) doubly confluent Heun Painlevé III′
(13) bi-confluent Heun Painlevé IV-34
(4) tri-confluent Heun Painlevé II–I
The main topic of our paper, described in Section 4, is a derivation of Painlevé equations
from Heun class equations. The first step of this derivation is a choice of a family of Heun
class equations depending on a parameter denoted t and called the time. Then we consider
then the corresponding deformed Heun class equation, (1.2), which depends on two additional
variables: λ, µ. The conditions for a constant monodromy lead to a set of nonlinear differential
equation for λ, µ in terms of t. These equations can be interpreted as Hamilton equations
generated by H(t, λ, µ), which we will call Painlevé Hamiltonians.
The most difficult ingredient of the passage from Heun class to Painlevé is the choice of the
time variable and of the so-called compatibility functions a, b, which control the isomonodromic
deformations. The main idea of our paper is to present this passage in a unified way. Our first
attempt in this direction is described in Theorem 4.1. The description of compatibility functions
and the corresponding Hamiltonian contained in this theorem is unified, however it is not very
transparent. Theorem 4.2, which can be viewed as the first part of the main result of this paper,
is more satisfactory. The method of isomodronomic deformation is here divided into two slightly
different ansatzes, called Cases A and B. The main condition on σ, τ , η, t for the applicability
of Ansatz A is the existence of a zero of σ, so that we can write σ(z) = (z − s)ρ(z), where ρ is
a polynomial of degree ≤ 2. Under these conditions there exists a function t 7→ m(t) such that
H(t, λ, µ) = m
(
σ(λ)µ2 + (τ(λ)− (λ− s)ρ′(λ))µ+ η(λ)
)
(1.3)
is a Painlevé Hamiltonian.
Among the conditions on σ, τ , η, t needed for Ansatz B the most important is deg σ ≤ 2.
Then we can show that there exists t 7→ m(t) such that
H(t, λ, µ) = m
(
σ(λ)µ2 + (τ(λ)− σ′(λ))µ+ η(λ)
)
(1.4)
is a Painlevé Hamiltonian.
It seems impossible to implement Theorem 4.2 in a fully unified way, and one has to subdivide
Cases A and B into several subcases. Case A splits into Subcases A1, Ap, Aq and Case B splits
into Subcases Bp and Bq. We describe these subcases in Theorem 4.3. The main difference
between the subcases is the choice of the time variable t:
In Subcase A1 the variable t is the position of one of Fuchsian singularities.
In Subcases Ap and Bp the variable t is contained in τ .
In Subcases Aq and Bq the variable t is contained in η.
In the following list we informally describe when we can apply various subcases.
A1. σ(t) = 0, σ′(t) 6= 0, deg(z − t)η ≤ 2.
Ap. σ(s) = σ′(s) = 0, τ(s) 6= 0, ση(s) = (ση)′(s) = 0.
6 J. Dereziński, A. Ishkhanyan and A. Latosiński
Aq. σ(s) = σ′(s) = 0, τ(s) = 0, ση(s) = 0, (ση)′(s) 6= 0.
Bp. deg σ ≤ 2, deg τ = 2, deg ση ≤ 2.
Bq. deg σ ≤ 2, deg τ ≤ 1, deg ση = 3.
Note that Subcase A1 works in the standard Heun type (1111), leading to Painlevé VI. Hence
it works in under generic conditions. However, it does not works for some confluent types.
For instance, for most degenerate types one needs to use either Subcase Aq or Subcase Bq.
Altogether, Theorem 4.3 works for certain normal forms of all types of Heun class equations.
This allows us to derive all types of Painlevé equations.
The derivation of the Painlevé VI equation from the Heun equation can be traced back to
a paper by Fuchs [6] from the early 20th century (written by the son of Fuchs from whose
name the adjective “Fuchsian” comes). The approach was generalized to other Painlevé equa-
tions by Okamoto [20] and refined by Ohyama–Okumura [19]. A discussion of the relationship
between the biconfluent Heun type and the Painlevé IV equation can be also found in [1].
Thus derivations described in Sections 4.5–4.15 are known. They are in particular described
by Ohyama–Okumura [19], where they were checked case by case. Our approach allows us to
automatize these derivations and view them as implementations of a unified algorithm. In fact,
our paper can be to some extent viewed as an explanation of the principles that underly the
results of [19].
Slavyanov and Lay devote in [25] a whole chapter to the Heun class — Painlevé correspon-
dence. In particular, they stress that Painlevé Hamiltonians can be viewed as “dequantizations”
of the corresponding Heun class equations. In fact, the symbol of Heun class operators, that
is the expression obtained from (1.1) by replacing ∂z with µ and z with λ, is very similar
to (1.3) and (1.4). (The difference is a “lower order term”, which can be interpreted as a result
of an “ordering prescription”.) Nevertheless, to our understanding, [25] contains only a sketch
of a program. The details of this program are quite involved and [25] does not provide its full
description.
Clearly, every 2nd order scalar equation can be rewritten as a system of two 1st order equa-
tions. For instance, we can rewrite (1.1) as
σ(z)∂z
[
u(z)
v(z)
]
=
[
0 1
−σ(z)η(z) −τ(z) + σ′(z)
] [
u(z)
v(z)
]
(or in other ways). In particular, Heun class equations are essentially equivalent to certain
systems of 1st order equations called sometimes Heun connections. This suggests a different
approach to deriving Painlevé equations starting from Heun connections. This alternative ap-
proach was studied, e.g., by Jimbo and Miwa [13, 14]. A recent exposition of this approach can
be found in [11].
Note that in the above references each type of Painlevé equations were treated separately.
Besides, degenerate cases were usually left out. It would be interesting to investigate to what
extent a derivation of all types Painlevé equations from Heun connections can be treated in
a uniform way, in the spirit of Theorems 4.2 and 4.3. Anyway, in this paper we do not consider
this question and we stick to (scalar 2nd order) deformed Heun class equations.
2 Second order linear differential equations
in the complex domain
2.1 Differential equation and operator
Let us recall basic concepts of ordinary 2nd order linear differential equations in the complex
domain with holomorphic coefficients. They have the form (1.1), where σ, τ and η are holo-
From Heun Class Equations to Painlevé Equations 7
morphic functions. We will often describe the equation (1.1) by specifying the corresponding
operator
σ(z)∂2z + τ(z)∂z + η(z). (2.1)
Clearly, by multiplying an equation from the left by an arbitrary nonzero holomorphic func-
tion we obtain an equivalent equation. However, we change the corresponding operator. Speak-
ing of operators instead of equations, which we will often do, has two advantages. First it
saves a little space, since we do not need to write the function u. Besides, an operator contains
more information than the corresponding equation, therefore sometimes allows for making more
precise statements.
Divide (1.1) and (2.1) by σ(z) and set
p(z) :=
τ(z)
σ(z)
, q(z) :=
η(z)
σ(z)
.
This leads to the so-called principal form of the equation and the corresponding principal oper-
ator: (
∂2z + p(z)∂z + q(z)
)
u(z) = 0, (2.2)
A := ∂2z + p(z)∂z + q(z). (2.3)
The principal form is often used as the standard form. However, we will often prefer different
forms.
2.2 Singularities of functions
Let p be a function holomorphic on an open subset of the Riemann sphere C ∪ {∞}. Let
z0 ∈ C ∪ {∞} be its singularity, so that
p(z) =
∞∑
k=−∞
pz0,k(z − z0)k, |z − z0| < r for some r > 0, if z0 ∈ C,
p(z) =
∞∑
k=−∞
p∞,kz
k, |z| > R for some R ≥ 0, if z0 =∞.
We define the degree of the singularity of p at z0 by
deg(p, z0) := −min{k | pz0,k 6= 0}, z0 ∈ C,
deg(p,∞) := max{k | p∞,k 6= 0}, z0 =∞.
(If p is a polynomial, then deg(p,∞) = deg(p) is its usual degree.)
Note that if φ is a biholomorphic transformation of a neighborhood of z0 onto a neighborhood
of φ(z0), then
deg(p, z0) = deg
(
p ◦ φ−1, φ(z0)
)
.
2.3 Singularities of equations
Consider now the equation (2.2) (in the principal form) with holomorphic coefficients represented
by the operator (2.3), which we denote by A. We say that z0 is a regular point of A if
deg(p, z0) ≤ 0, deg(q, z0) ≤ 0, z0 ∈ C,
deg
(
p− 2
z
,∞
)
≤ −2, deg(q,∞) ≤ −4, z0 =∞.
Otherwise we say that it is a singular point of A.
8 J. Dereziński, A. Ishkhanyan and A. Latosiński
We say that the singular point z0 is regular or Fuchsian if
deg(p, z0) ≤ 1, deg(q, z0) ≤ 2, z0 ∈ C,
deg(p,∞) ≤ −1, deg(q,∞) ≤ −2, z0 =∞.
It is standard to introduce two indices of a Fuchsian singular point:
the roots of λ(λ− 1) + pz0,−1λ+ qz0,−2 are called indices of z0 ∈ C, (2.4)
the roots of λ(λ+ 1)− p∞,−1λ+ q∞,−2 are called indices of z0 =∞. (2.5)
The rank of A at z0 is defined as follows. If z0 is a regular point, we set rk(A, z0) = 0. The
case of rank equal to 1
2 is somewhat special:
rk(A, z0) :=
1
2
if deg
(
p− 1
2(z − z0)
, z0
)
≤ 0, deg(q, z0) ≤ 1, z0 ∈ C,
rk(A,∞) :=
1
2
if deg
(
p− 3
2z
,∞
)
≤ −2, deg(q,∞) ≤ −3, z0 =∞.
If rk(A, z0) 6= 0, 12 , then we set
rk(A, z0) := max
{
deg(p, z0),
1
2
deg(q, z0), 1
}
, z0 ∈ C,
rk(A,∞) := max
{
deg(p,∞) + 2,
1
2
deg(q,∞) + 2, 1
}
, z0 =∞.
The rank and indices of a singularity are invariants of biholomorphic transformations. For
instance, this is the case of homographies, that is w = az+b
cz+d , or z = dw−b
−cw+a , where ad − bc = 1.
We obtain
∂2z + p(z)∂z + q(z) = (−cw + a)4∂2w
+
(
−2c(−cw + a)3+ p
(
dw − b
−cw+ a
)
(−cw+ a)2
)
∂w + q
(
dw − b
−cw+ a
)
. (2.6)
In order to obtain the principal form we need to divide (2.3) by (−cw + a)4.
Note that the rank is always an integer or a half-integer. A singularity of rank m − 1
2 with
m ∈ {1, 2, . . . } can be often treated as a degeneration of a singularity of rank m. This motivates
us to introduce the rounded rank, denoted drke:
drke(A, z0) :=
{
rk(A, z0), if rk(A, z0) ∈ {0, 1, 2, . . . },
rk(A, z0) + 1
2 , if rk(A, z0) ∈
{
1
2 ,
3
2 ,
5
2 , . . .
}
.
Equivalently,
drke(A, z0) :=
⌈
max
{
deg(p, z0),
1
2
deg(q, z0), 0
}⌉
, z0 ∈ C,
drke(A,∞) :=
⌈
max
{
deg(p,∞) + 2,
1
2
deg(q,∞) + 2, 0
}⌉
, z0 =∞.
Here d·e is the ceiling function, that is
dxe := inf{n ∈ Z | x ≤ n}.
The singularity z0 is Fuchsian if its rank is 1
2 or 1. Thus z0 is a Fuchsian singularity iff its
rounded rank is 1.
According to our definition, a Fuchsian singularity has rank 1
2 if its indices are 0, 1
2 . The
splitting of the Fuchsian case into two subcases, the rank 1
2 and 1, is quite useful, even if its
definition is not obvious. Nevertheless, in our paper we will not make much use of this splitting
and both will be usually treated as one case, denoted 1.
From Heun Class Equations to Painlevé Equations 9
2.4 Sandwiching with functions
The family of equations (2.2) is preserved by several kinds of transformations of the form
e−r(z)
(
∂2z+ p(z)∂z+ q(z)
)
er(z) = ∂2z+
(
p(z)+ 2r′(z)
)
∂z+ q(z)+ p(z)r′(z)+ r′(z)2+ r′′(z)
=: ∂2z + p̃(z)∂z + q̃(z) = Ã. (2.7)
Sandwiching with powers. For r(z) = κ log(z)
(z − z0)−κ
(
∂2z + p(z)∂z + q(z)
)
(z − z0)κ = ∂2z +
(
p(z) + 2κ(z − z0)−1
)
∂z + q(z)
+ p(z)κ(z − z0)−1 + (κ2 − κ)(z − z0)−2. (2.8)
If z0 ∈ C ∪ {∞} is a singularity of (2.2) and rk(z0) > 1, then the transformation (2.8) preserves
its rank. If z0 is a Fuchsian singularity, then after the transformation it is also Fuchsian.
If ρ1, ρ2 are the indices of (2.2) at z0 ∈ C, then ρ1 + κ, ρ2 + κ are the indices of (2.8) at z0.
The same is true for ∞, except that the indices of (2.8) at ∞ are ρ1 − κ, ρ2 − κ.
Sandwiching with exponentials. Let k = 2, 3, . . . . We have
exp
(
κ(z − z0)−k+1
k − 1
)(
∂2z + p(z)∂z + q(z)
)
exp
(
−κ(z − z0)−k+1
k − 1
)
= ∂2z +
(
p(z) + 2κ(z − z0)−k
)
∂z + q(z) + p(z)κ(z − z0)−k
+ κ2(z − z0)−2k − κk(z − z0)−k−1. (2.9)
Hence this transformation preserves rk(z0) if it is > k, and preserves or decreases rk(z0) if
it is = k. The transformation does not change the coefficients pz0,−1, qz0,−1, qz0,−2. Therefore,
it also does not change the indices of z0.
The same is true for ∞ under the transformation
exp
(
−κz
k+1
k + 1
)(
∂2z + p(z)∂z + q(z)
)
exp
(
κzk+1
k + 1
)
= ∂2z +
(
p(z) + 2κzk
)
∂z + q(z) + p(z)κzk + κ2z2k + κkzk−1. (2.10)
More generally, we have transformations of the form (2.7), where
r(z) =
−1∑
k=−m+1
wk
k
(z − z0)k + w0 log(z − z0), (2.11)
so that r′(z) =
0∑
k=−m+1
wk(z − z0)k−1 if z0 ∈ C;
r(z) = −
m−1∑
k=1
wk
k
zk − w0 log(z), (2.12)
so that r′(z) = −
m−1∑
k=0
wkz
k−1 if z0 =∞.
We define the absolute rank of A at z0 as
Rk(A, z0) := inf
{
rk(Ã, z0)
}
,
where à are all possible transforms of A of the form (2.7) with r as in (2.11) or (2.12).
10 J. Dereziński, A. Ishkhanyan and A. Latosiński
2.5 Half-integer rank
In this subsection we discuss singular points with a half-integer rank. They are in a sense
exceptional and have special properties.
Suppose that the equation (2.2) has a singular point at z0 and rk(A, z0) = m + 1
2 , where
m = 0, 1, . . . . It is easy to see that this implies
Rk(A, z0) = rk(A, z0).
Without loss of generality we can assume that the singularity is at 0. This is equivalent to
deg(p, 0) ≤ m, deg(q, 0) = 2m+ 1, m ≥ 1,
deg
(
p− 1
2z
, 0
)
≤ 0, deg(q, 0) ≤ 1, m = 0.
Let us make the substitution
z = y2, y =
√
z. (2.13)
Using ∂z = 1
2y∂y we transform (2.3) into
1
4y2
∂2y −
1
4y3
∂y +
p(y2)
2y
∂y + q
(
y2
)
. (2.14)
Multiplying (2.14) by 4y2 we obtain an equation in the principal form
∂2y + 2y
(
p
(
y2
)
− 1
2y2
)
∂y + 4y2q
(
y2
)
. (2.15)
Now
deg
(
2y
(
p
(
y2
)
− 1
2y2
)
, 0
)
≤ 2m− 1,
deg
(
4y2q
(
y2
)
, 0
)
= 2(2m+ 1)− 2 = 4m, m ≥ 1,
deg
(
4y2q
(
y2
)
, 0
)
≤ 0, m = 0.
Thus the rank of (2.15) at zero is 2m.
Thus we have shown that by a quadratic substitution we can reduce a singularity of a half-
integer rank to a singularity of integer rank. The resulting equation (2.15) is in addition invariant
with respect to the substitution y → −y and the rank of the singularity is even.
Note that our definition of rank 1
2 has been chosen so that the above quadratic reduction
works for all half-integer ranks.
2.6 Simplifying the equation
Suppose that the equation (2.2) has a singular point at z0. Obviously, we have 3 exclusive
possibilities:
(0) Rk(A, z0) ≤ 1;
(1) Rk(A, z0) is an integer and ≥ 2;
(2) Rk(A, z0) is a half-integer and ≥ 3
2 .
We would like to simplify the equation around this singularity by sandwiching with er, where r
is given by (2.11) or (2.12). The transformed operator will be, as usual, denoted Ã. We will see
that the simplification will be quite different depending on Case (1) and (2). (Case (0) is simple
enough, therefore we do not discuss it in the following proposition).
From Heun Class Equations to Painlevé Equations 11
Proposition 2.1.
(1) If Rk(A, z0) is an integer and ≥ 2, then there exist exactly two transformations such that
Rk(A, z0) = rk(Ã, z0) = deg(p̃, z0) ≥ deg(q̃, z0), z0 ∈ C,
Rk(A,∞) = rk(Ã,∞) = deg(p̃,∞) + 2 ≥ deg(q̃,∞) + 4, z0 =∞. (2.16)
(2) If Rk(A, z0) is a half-integer and ≥ 3
2 , then there exists a unique transformation such that
deg(p̃, z0) ≤ 0 and Rk(A, z0) = rk(Ã, z0) =
1
2
deg(q̃, z0), z0 ∈ C,
deg(p̃,∞) ≤ −2 and Rk(A,∞) = rk(Ã,∞) =
1
2
deg(q̃,∞) + 2, z0 =∞. (2.17)
Proof. Without loss of generality we can assume that the singularity is at 0. We will use the
identity
q̃(z) =
∞∑
n=−2m
zn
(
min(−1,n+m)∑
k=max(−m,n+1)
wk+1wn−k+1 + (n+ 1)wn+2 +
min(−1,n+m)∑
k=−m
wk+1pn−k + qn
)
. (2.18)
We will apply one of the following three transformations, denoted I, II and III.
Transformation I. Suppose that the rank of the initial equation is m− 1
2 , m = 2, 3, . . . . Then
deg(q, 0) = 2m− 1 and
p(z) =
∞∑
j=−m+1
pjz
j .
We choose r(z) such that
r′(z) = −1
2
−1∑
j=−m+1
pjz
j .
Then deg(p̃, 0) ≤ 0 and deg(q̃, 0) = deg(q, 0). The transformed equation satisfies (2.17).
For transformations II and III we suppose that the rank of the initial equation is m = 2, 3, . . . .
Transformation II. Assume that
p2−m 6= 4q−2m.
Let w−m+1 be one of two solutions of
0 = w2
−m+1 + w−m+1p−m + q−2m. (2.19)
Then q̃−2m = 0. Equating q̃−m+j = 0 for j = −m + 1, . . . ,−1 we obtain from (2.18) the
recurrence relations
0 = wj+1(2w−m+1 + p−m)
+
j−1∑
k=−m+1
wk+1w−m+j−k+1 + (−m+ j + 1)w−m+j+2 +
j−1∑
k=−m
wk+1p−m+j−k + q−m+j .
Using
2w−m+1 + p−m =
√
p2−m − 4q−2m 6= 0
we can solve the recurrence relations. The transformed equation satisfies (2.16).
12 J. Dereziński, A. Ishkhanyan and A. Latosiński
Transformation III. If
p2−m = 4q−2m,
then we sandwich with er, where r(z) = p−mz−m+1
2(−m+1) . The transformed operator à has deg(p̃, 0) ≤
m− 1 and deg(q̃, 0) ≤ 2m− 1. Thus after the transformation rk
(
0, Ã
)
≤ m− 1
2 . If the resulting
rank is half-integer, then we apply I and stop. If the resulting rank is an integer, we apply II
and stop, or III and we iterate.
We have thus two possibilities:
Case (1) First a finite number of III, and then II. At the end we obtain equation satisfying (2.16).
Steps III are uniquely determined and II has two possibilities.
Case (2) First a finite number of III, and then I obtaining (2.17). All steps are uniquely deter-
mined. �
We will say that the operator A has a grounded form at z0 if deg(p, z0) ≥ deg(q, z0). If z0 is
Fuchsian, then this is equivalent to one of the indices being 0.
It follows from Proposition 2.1 that if Rk(A, z0) is an integer or 1
2 , then the equation can be
brought to a grounded form at z0.
2.7 Solutions in terms of formal power series
We consider the equation (2.2) and try to solve it in terms of a nontrivial, not necessarily
convergent power series
∞∑
k=0
vkz
k.
As we will see, this is not always possible.
Proposition 2.2. Set m := deg(p, 0) and l := deg(q, 0).
1. If m ≤ 0 and l ≤ 0, then for any v0, v1 there exists a power series solution.
2. If m ≤ 1 and l ≤ 2, then there are no power series solutions unless for some n = 1, 2, . . .
we have
n(n− 1) + np−1 + q−2 = 0. (2.20)
3. If m ≥ 2 and l ≤ m, then for any v0 there is a unique power series solution.
4. If m ≥ 2 and l = m+1, then there is no power series solution unless for some n = 1, 2, . . .
p−mn+ q−m−1 = 0.
5. If m ≥ 1 and l ≥ m+ 2, then there are no power series solutions.
Proof. By equating the terms at zn to zero in(
∂2z +
∞∑
j=−m
pjz
j∂z +
∞∑
j=−l
qjz
j
) ∞∑
k=0
vkz
k = 0
we obtain the following equations:
(n+ 2)(n+ 1)vn+2 +
n+1+m∑
k=1
pn+1−kkvk +
n+l∑
k=0
qn−kvk = 0, n ∈ Z.
From Heun Class Equations to Painlevé Equations 13
Let us prove (5). For j := n+ l = 0, . . . , l −m− 1 we obtain the equations
0 = q−lv0,
· · · · · · · · · · · · · · · · · · · · · · · ·
0 = q−lvj + · · ·+ q−l+jv0,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
0 = q−lvl−m−1 + · · ·+ q−m−1v0.
Thus 0 = v0 = · · · = vl−m−1.
If j ≥ l −m, there are additional terms coming from the 1st order derivative. vk with the
highest k contained in such a term has k = m − l + j + 1. But because of l ≥ m + 2 we have
k ≤ j − 1. Thus vk = 0 by one of previous recursion steps.
If j ≥ l, there is an additional term coming from the 2nd order derivative, involving vk with
k = −l + j + 2. But l ≥ 3 implies k ≤ j − 1. Again, this vk = 0 by one of previous recursion
steps.
Let us prove (4). We have recursion relations
0 = q−lv0,
0 = (p−l+1 + q−l)v1 + q−l+1v0,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
0 = (jp−l+1 + q−l)vj + · · · ,
where dots denote terms depending on vj−1, . . . , v0. If (2.20) has no solutions, then we can solve
the recurrence obtaining 0 = v0 = v1 = · · · .
Let us prove (3). We have the recursion relations
0 = p−mv1 + q−lv0,
0 = p−m2v2 + p−m+1v1 + q−mv1 + q−m+1v0, (2.21)
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
0 = jp−mvj + · · · ,
where · · · involve v0, . . . , vj−1. If m = 2, there is an additional term 2v0 in (2.21). Note that
p−m 6= 0. Hence, for any v0 we can solve the recurrence obtaining v1, v2, . . . .
(2) follows immediately from the well-known theory of solutions around a Fuchsian singular
point.
(1) is the well-known fact about the Cauchy problem in the regular case. �
2.8 Thomé solutions
By the so-called Frobenius method, if z0 is a Fuchsian singularity and ρ1, ρ2 are its indices such
that ρ1 − ρ2 6∈ Z, then solutions of the equation (2.2) are spanned by two convergent power
series indexed by i = 1, 2 with vi,0 6= 0:
∞∑
j=0
vi,j(z − z0)j+ρi , z0 ∈ C,
0∑
j=−∞
vi,jz
j−ρi , z0 =∞.
14 J. Dereziński, A. Ishkhanyan and A. Latosiński
If ρ1 − ρ2 ∈ Z this is not always true. One can then assume that ρ1 − ρ2 ≥ 0. There exists one
solution as above with i = 1 and the second has the form
∞∑
j=0
v2,j(z − z0)ρ2+j + log z
∞∑
j=0
v1,j(z − z0)ρ1+j , z0 ∈ C,
0∑
j=−∞
v2,jz
−ρ2+j + log z
0∑
j=−∞
v1,jz
−ρ1+j , z0 =∞.
If the singular point is not Fuchsian, then we can also look for solutions in a similar form,
however the resulting power series are usually no longer convergent. One obtains the so-called
Thomé solutions. Note that in some way the situation is simpler, because we do not have the
logarithmic case. On the other hand, half-integer ranks need to be treated separately and lead
to power series in
√
z.
Proposition 2.3. Let z0 be a singular point of A with rk(A, z0) = n. Then there exist two
formal solutions of A, indexed by i = 1, 2.
1. If Rk(A, z0) is an integer ≥ 2, they have the form
exp
( −1∑
k=−n+1
(z − z0)k
k
wi,k
) ∞∑
j=0
vi,j(z − z0)wi,0+j , z0 ∈ C,
exp
(
−
n−1∑
k=1
zk
k
wi,k
)
0∑
j=−∞
vi,jz
−wi,0+j , z0 =∞.
2. Let Rk(A, z0) be a half integer ≥ 3
2 . The two formal solutions have the form
exp
( − 1
2∑′
k=−n+1
(z − z0)k
k
wi,k
) ∞∑′
j=0
vi,j(z − z0)wi,0+j , z0 ∈ C,
exp
(
−
n−1∑′
k= 1
2
zk
k
wi,k
) 0∑′
j=−∞
vi,jz
−wi,0+j , z0 =∞.
Here
∑′ denotes the sum where the index k within its range runs over both integers and
half-integers. We have
w0,k = (−1)2kw1,k, v0,k = (−1)2kv1,k. (2.22)
Proof. For simplicity, assume that z0 = 0.
Suppose that Rk(A, 0) is an integer ≥ 2. By Proposition 2.1(1) we can transform the equation
to a grounded form in two distinct ways. By Proposition 2.2(3), the grounded form has a solution
in terms of the power series.
If Rk(A, 0) = m + 1
2 is a half-integer ≥ 3
2 , first we reduce the equation to the form with
a half-integer rank, see Proposition 2.1(2). Then we apply the quadratic transformation, as
described in Section 2.5. We obtain an even equation in
√
z of the rank 2m, with p̃−2m = 0 and
q̃−4m 6= 0. We already know that it has a solution of the form described in 1:
exp
( −1∑
k=−2m+1
(√
z
)k
k
w̃i,k
) ∞∑
j=0
ṽj
(√
z
)w̃i,0+j . (2.23)
From Heun Class Equations to Painlevé Equations 15
Using 2m ≥ 2 and (2.19) we obtain
w̃−2m+1 =
√
−q̃−4m 6= 0. (2.24)
Clearly, (2.23) with
√
z replaced by −
√
z is also a solution. By (2.24) both solutions are not
proportional to one another. This proves (2.3). �
Note that Proposition 2.3 is also true in the Fuchsian case, except that for Rk(A, z0) = 1,
one has to make an obvious modification in the logarithmic case, and for Rk(A, z0) = 1
2 (2.3)
does not have to be true.
In the Fuchsian case wi,0, i = 1, 2, coincide with the indices of z0. In what follows, the
numbers wi,0, i = 1, 2, will be called indices of z0 in the general case as well. We also introduce
the alternative notation
ρz0,i := wi,0, i = 1, 2. (2.25)
Proposition 2.4. Let z0 be a singularity of A.
1. We have
ρz0,1 + ρz0,2 = −pz0,−1 + Rk(A, z0), z0 ∈ C, (2.26)
ρ∞,1 + ρ∞,2 = p∞,−1 − 2 + Rk(A,∞), z0 =∞. (2.27)
2. If Rk(A, z0) ∈
{
3
2 ,
5
2 , . . .
}
, then
ρz0,1 = ρz0,2 =
1
2
(
−pz0,−1 + Rk(A, z0)
)
, z0 ∈ C,
ρ∞,1 = ρ∞,2 =
1
2
(
p∞,−1 − 2 + Rk(A,∞)
)
, z0 =∞.
3. If z0 ∈ C is grounded, then
{ρz0,1, ρz0,2} =
{
0,−pz0,−1 + Rk(A, z0)
}
.
Proof. (1) Without loss of generality we can assume that z0 = 0. When we apply sandwich-
ing with er, where r(z) has the form (2.11), then p−1 and ρi are transformed into p−1 − 2w0
and ρi + w0. This does not affect the identity (2.26).
Assume first that Rk(A, 0) is an integer. Then by a sandwiching transformation we can
reduce the equation to a grounded form at 0. The first m recurrence relations of (2.18) read
then
0 =
n+m∑
k=−m
wk+1(wn−k+1 + pn−k), n = −2m, . . . ,−m− 2,
0 =
−1∑
k=−m
wk+1(w−m−k + p−m−1−k)−mw−m+1.
Apart from the solution wk+1 = 0, k = −m, . . . ,−1, this is solved by
wk+1 := −pk, k = −m, . . . ,−2,
w0 := −p−1 +m.
Thus {ρ1, ρ2} = {0,−p−1 +m}. Hence (2.26) is satisfied.
16 J. Dereziński, A. Ishkhanyan and A. Latosiński
Assume next that Rk(A, 0) is a half integer equal m+ 1
2 . After an appropriate sandwiching
transformation we can assume that Rk(A, 0) = rk(A, 0). Then we can apply the quadratic
transformation (2.13) obtaining an equation
à = ∂2y + p̃(y)∂y + q̃(y).
Let ρ̃i, i = 1, 2 be the indices of à at zero. Now rk
(
Ã, 0
)
= 2m, which is an integer. Hence we
can apply the formula (2.26)
ρ̃1 + ρ̃2 = −p̃−1 + 2m.
But the indices of A at 0 are ρi = ρ̃i
2 , i = 1, 2 and p̃−1 = 2p−1 − 1. Hence
ρ1 + ρ2 =
1
2
(ρ̃1 + ρ̃2) =
1
2
(−2p−1 + 1 + 2m) = −p−1 +m+
1
2
.
This ends the proof (2.26).
To prove (2.27) we apply the transformation w = 1
z . We note that Rk(A,∞) is transformed
to Rk(A, 0) and ρ∞,i are transformed to ρ0,i, i = 1, 2. Besides, p∞,−1 is transformed to −p0,−1+2.
(1) and (2.3) imply (2).
(1) and (2.2) of Proposition 2.2 imply (3). �
2.9 Nonlogarithmic singularities
Let z0 be a singular point of the equation (2.2). We say that z0 is nonlogarithmic or apparent iff
all solutions of the equation are meromorphic around this singularity. If z0 is a nonlogarithmic
Fuchsian singular point, then both its indices are integers.
The following proposition shows how to deform a given equation so that one obtains an addi-
tional nonlogarithmic singularity with indices 0, 2. This deformation depends on two parame-
ters λ, µ: the additional singularity is located at λ, and solutions of the deformed equation
satisfy µv(λ) = v′(λ). It will play the central role in the derivation of Painlevé equations from
Heun class equations in Section 4.
Proposition 2.5. Let σ, τ , η be analytic at λ and σ(λ) 6= 0. Let µ ∈ C. Then all solution of
the equation given by
σ(z)∂2z +
(
τ(z)− σ(z)
z − λ
)
∂z + η(z)− η(λ)− µ2σ(λ)− µ
(
τ(λ)− σ′(λ)
)
+
µσ(λ)
z − λ
(2.28)
are analytic at λ. Thus the equation given by (2.28) has a nonlogarithmic singularity at z = λ.
The singularity is Fuchsian with indices 0, 2.
Proof. We look for a solution analytic around λ:
v(z) =
∞∑
n=0
vn(z − λ)n.
We obtain(
−σ(λ)v1 + µσ(λ)v0
)
(z − λ)0 +
(
σ(λ)2v2 +
(
τ(λ)− σ′(λ)
)
v1 − σ(λ)2v2 −
(
µτ(λ)
− µσ′(λ) + µ2σ(λ)
)
v0 + µσ(λ)v1
)
(z − λ)1
+
∞∑
n=3
(
σ(λ)n(n− 2)vn + · · ·
)
(z − λ)n−1 = 0.
We have σ(λ) 6= 0. Hence the first line implies v1 = µv0. Then the second line is identically
zero, and v2 is left unspecified. The next terms yield recurrence relations for vn, n = 3, . . . . �
From Heun Class Equations to Painlevé Equations 17
3 Equations with rational coefficients
3.1 The Mn class and the grounded Mn class
Consider an equation given by the operator
A := ∂2z + p(z)∂z + q(z), (3.1)
where p(z), q(z) are rational functions.
If z1, . . . , zk ∈ C ∪ {∞} are its singularities and their ranks are m1, . . . ,mk, then we will say
that the equation (3.1) is of type (m1m2 · · ·mk)
Often we will need a more precise description of (3.1), which gives information what is the
rank of the singularity at ∞. We will then put it at the end of the sequence, so that zk = ∞,
and precede it with a semicolon. We will write that (3.1) is of type (m1m2 · · ·mk−1;m∞).
By writing mi instead of mi we will mean the rounded rank. We will use it especially often
for 1. Thus 1 means a Fuchsian singularity
(
1 or 1
2
)
.
Every equation having no more than n+1 singular points in the Riemann sphere, all of them
Fuchsian and at most n finite, is given by an operator of the form
∂2z +
n∑
j=1
aj
z − zj
∂z +
n∑
j=1
bj
z − zj
+
n∑
j=1
cj
(z − zj)2
with
n∑
j=1
bj = 0, (3.2)
where z1, . . . , zn are distinct points in C. The family of equations (3.2) will be called the Mn
type. The corresponding symbol is
(
1 · · · 1
n−1 times
; 1
)
.
Each finite singularity has at least one index equal 0 if and only if c1 = · · · = cn−1 = 0. Thus
such equations are given by operators
∂2z +
n∑
j=1
aj
z − zj
∂z +
n∑
j=1
bj
z − zj
with
n∑
j=1
bj = 0. (3.3)
The family of equations given by (3.3) will be called the grounded Mn type.
Proposition 3.1. By sandwiching with powers, as in (2.8), we can always transform an Mn
type equation into a grounded Mn type equation.
We say that a differential equation belongs to the Mn class if it is given by
∂2z +
τ(z)
σ(z)
∂z +
ξ(z)
σ(z)2
, (3.4)
where σ, τ , ξ are polynomials satisfying
σ 6= 0, deg σ ≤ n, deg τ ≤ n− 1, deg ξ ≤ 2n− 2. (3.5)
We will often use the shorthand
η(z) :=
ξ(z)
σ(z)
,
where η does not have to be a polynomial.
We say that a differential equation belongs to the grounded Mn class if it is given by
∂2z +
τ(z)
σ(z)
∂z +
η(z)
σ(z)
, (3.6)
where σ, τ , η are polynomials satisfying
σ 6= 0, deg σ ≤ n, deg τ ≤ n− 1, deg η ≤ n− 2.
The name “the Mn class” is borrowed from Lay–Slavyanov [25].
18 J. Dereziński, A. Ishkhanyan and A. Latosiński
Proposition 3.2.
1. The Mn type is contained in the Mn class. An equation of the Mn class is of the Mn type
iff σ possesses n distinct roots.
2. The grounded Mn type is contained in the grounded Mn class. An equation of the grounded
Mn class is of the grounded Mn type iff σ possesses n distinct roots.
Proof. Let us prove (1). Consider (3.2). Set
σ(z) := (z − z1) · · · (z − zn).
Then clearly σ is a nonzero polynomial with n distinct roots. We easily see that (3.2) can be
rewritten as (3.4) with (3.5) satisfied.
Conversely, consider (3.4) such that σ has n distinct roots, namely, z1, . . . , zn. Then we can
decompose τ(z)
σ(z) and ξ(z)
σ(z)2
into simple fractions, obtaining (3.2).
The proof of (2) is analogous. �
Proposition 3.3. Let (z1, . . . , zk) be the singularities of an equation A of the Mn class. Then
drke(A, z1) + · · ·+ drke(A, zk) ≤ n+ 1.
Proof. Without loss of generality we can assume that z1, . . . , zk−1 ∈ C and zk =∞. Let
σ(z) = (z − z1)m1 · · · (z − zk−1)mk−1
with distinct zi’s. Then z1, . . . , zk−1 are the finite singular points. Clearly, drke(A, zi) ≤ mi.
Therefore,
drke(A, z1) + · · ·+ drke(A, zk−1) ≤ deg σ. (3.7)
Now
drke(A,∞) =
⌈
max
(
deg τ − deg σ + 2,
1
2
(deg ξ − 2 deg σ) + 2
)⌉
≤
⌈
max
(
n− 1− deg σ + 2,
1
2
(2n− 2− 2 deg σ) + 2
)⌉
= n+ 1− deg σ. (3.8)
Then we sum (3.7) and (3.8). �
We will often represent Mn class equations by operators obtained by multiplying (3.4) or (3.6)
from the right by σ(z):
σ(z)∂2z + τ(z)∂z + η(z). (3.9)
Obviously, Mn class equations and operators are defined by coefficients of the polynomials σ,
τ , ξ. Therefore, they form a complex manifold parameterized by(
Cn\{0}
)
× C3n−3. (3.10)
The condition saying that σ has n distinct roots defines an open dense subset in (3.10). Thus
the Mn type is an open dense subset of the Mn class. Hence the Mn class consists of the Mn
type and its limiting points in the topology of (3.10). These limiting points are traditionally
called confluent cases.
Similarly, grounded Mn class equations and operators are defined by σ, τ , η. Therefore, they
form a complex manifold parameterized by(
Cn\{0}
)
× C2n−3.
Clearly, the grounded Mn type is an open dense subset of the grounded Mn class. One can say
that the grounded Mn class consists of the grounded Mn type and its confluent cases.
From Heun Class Equations to Painlevé Equations 19
3.2 Generalized Fuchs relation
Recall that for any singular point z0 of an equation A we defined its two indices ρz0,1 and ρz0,2.
For Fuchsian singularities they were defined in (2.4), (2.5) and for non-Fuchsian singularities
in (2.25). If all singularities are regular then the well-known Fuchs relation says that the sum of
all indices equals the number of singularities minus 2. In the following proposition we describe
its generalization which is valid if some of the singularities are non-Fuchsian.
Proposition 3.4. Let z1, . . . , zk be the singular points of an equation A. Then
k∑
j=1
(ρzj ,1 + ρzj ,2) =
k∑
j=1
Rk(A, zj)− 2. (3.11)
Proof. Without loss of generality we can assume that zk =∞. We have
ρzi,1 + ρzi,2 = −pzj ,−1 + Rk(A, zj), j = 1, . . . , k − 1, (3.12)
ρ∞,1 + ρ∞,2 = p∞,−1 − 2 + Rk(A,∞), (3.13)
p∞,−1 =
k−1∑
j=1
pzj ,−1,
where (3.12) and (3.13) follows from Proposition 2.4. Summing up the above three relations we
obtain (3.11). �
3.3 Riemann class equations
The simplest nontrivial Mn class is the M2 class. We call it the Riemann class since it consists
of the Riemann equation with one singularity at∞ and its confluent cases. Thus Riemann class
operators have the form (3.4), where
σ 6= 0, deg σ ≤ 2, deg τ ≤ 1, deg ξ ≤ 2.
The grounded Riemann class operators has the form (3.6), where
σ 6= 0, deg σ ≤ 2, deg τ ≤ 1, η is a number.
Note that grounded Riemann class equations appear in the literature very often. They are
often called hypergeometric type equations, see [2, 4, 18].
It is well known that by a division by a constant, transformations z 7→ az + b, sandwiching
with powers and exponentials all Riemann class operators can be transformed into one of the
following types:
the 2F1 operator (11; 1) z(1− z)∂2z +
(
c− (a+ b+ 1)z
)
∂z − ab
the 2F0 operator (2; 1) z2∂2z +
(
−1 + (a+ b+ 1)z
)
∂z + ab
the 1F1 operator (1; 2) z∂2z + (c− z)∂z − a
the 0F1 operator (1; 3
2 ) z∂2z + c∂z − 1
the Hermite operator (; 3) ∂2z − 2z∂z − 2a
the Airy operator (; 5
2 ) ∂2z + z
the Euler II operator (1; 1) z2∂2z + cz∂z
the Euler I operator (1; 1) z∂2z + c∂z
the 1d Helmholtz operator (; 2) ∂2z + 1
the 1d Laplace operator (; 1) ∂2z
20 J. Dereziński, A. Ishkhanyan and A. Latosiński
Let us make some remarks.
1. The Euler II and Euler I operators yield the same equations.
2. The last four equations from the table can be solved in elementary functions.
3. In this table, only the Airy equation cannot be brought to the grounded form.
4. When we take into account the transformation z 7→ z−1, then the types (2; 1) and (1; 2)
are equivalent.
5. There are more relations between various types when we consider more complicated trans-
formations.
3.4 Heun class equations
M3 type equations were studied by Heun in [8]. Therefore, it is natural to call the M3 type the
Heun type. Consequently, the M3 class will be called the Heun class. The grounded M3 class
will be called the grounded Heun class.
Our terminology is consistent with [24, 25]. However, in some publications the name Heun
class is used to denote what we call the grounded Heun class, see, e.g., [5].
We will represent Heun class equations by Heun class operators. More precisely, we will say
that
σ(z)∂2z + τ(z)∂z + η(z) (3.14)
is a Heun class operator if η(z) = ξ(z)
σ(z) and σ, τ , ξ are polynomials such that
σ 6= 0, deg σ ≤ 3, deg τ ≤ 2, deg ξ ≤ 4. (3.15)
(3.14) is a grounded Heun class operator if σ, τ , η are polynomials such that
σ 6= 0, deg σ ≤ 3, deg τ ≤ 2, deg η ≤ 1.
If in addition σ has 3 distinct roots, then (3.14) is a (grounded) Heun type operator.
Clearly, the Heun class and the grounded Heun class are preserved by transformations z 7→
az + b.
The Heun class is also preserved by sandwiching with powers and exponentials, see (2.8), (2.9)
and (2.10).
Heun class operators are invariant with respect to swapping a finite singularity with the
infinity. More precisely, Heun class operators of the form (3.14) after the transformations
w = (z − z0)−1, where z0 is one of finite singular points
remain in the Heun class. Indeed, without loss of generality, we can suppose that z0 = 0. Thus
σ(0) = 0, so that σ(z) = zρ(z), where ρ is a polynomial with deg ρ ≤ 2. Substitute w = z−1,
which transforms (3.14) into
w3ρ
(
w−1
)
∂2w + w2
(
2ρ
(
w−1
)
− τ
(
w−1
))
∂w + η
(
w−1
)
. (3.16)
It is easy to see that
σ̃(w) := w3ρ
(
w−1
)
, (3.17a)
τ̃(w) := w2
(
2ρ
(
w−1
)
− τ
(
w−1
))
, (3.17b)
η̃(w) := η
(
w−1
)
(3.17c)
still satisfy the condition (3.15).
From Heun Class Equations to Painlevé Equations 21
Note that we do not need to put any prefactor in (3.16). Remarkably, the analogous property
does not hold for the Mn classes with n 6= 3: for them after swapping a finite singularity with∞
an additional prefactor is needed.
Swapping a finite singularity with ∞ is possible also if we want to stay within the grounded
Heun class, except that the transformation w = (z − z0)−1 needs to be followed by sandwiching
with a power, that is a transformation (2.8). Indeed, assume (3.9) is a grounded Heun class
operator and z0 = 0 is a singularity. Let α satisfy the generalized indicial equation at z =∞:
ρ′′
2
α(α+ 1)− τ ′′
2
α+ η′ = 0.
Then
w−α
(
σ̃(w)∂2w + τ̃(w)∂w + η̃(w)
)
wα = σ̃(w)∂2w + τ̃1(w)∂w + η̃1(w), (3.18)
where
τ̃1(w) := 2(α+ 1)w2ρ
(
w−1
)
− w2τ
(
w−1
)
,
η̃1(w) := wα
(
(α+ 1)ρ(0)− τ(0)
)
+ α
(
(α+ 1)ρ′(0)− τ ′(0)
)
+ η(0).
Clearly, (3.18) is a grounded Heun class operator.
3.5 Deformed Heun class equations
Consider σ, τ , η satisfying the conditions (3.15), so that (3.14) is a Heun class operator. Let
λ, µ ∈ C. The corresponding deformed Heun class operator is defined as
σ(z)∂2z +
(
τ(z)− σ(z)
z − λ
)
∂z + η(z)− η(λ)− σ(λ)µ2 −
(
τ(λ)− σ′(λ)
)
µ+
σ(λ)µ
z − λ
. (3.19)
By Proposition 2.5, the equation defined by (3.19) has a nonlogarithmic singularity at z = λ
with indices 0, 2. All the remaining finite singularities have the same type (the rank, the indices),
as for the original Heun class operator (3.14).
Thus to every Heun class operator (3.14) there corresponds a family of deformed Heun class
operators (3.19) depending on two new parameters: λ and µ. Note that one of the parameters
of η in the original operator (2.28) is lost — (3.19) does not depend on the free (zeroth order)
term of η.
The family of deformed Heun class operators is preserved by the same transformations as the
family of Heun class operators. Clearly, it is preserved by z 7→ az + b, division by a constant
and sandwiching with powers and exponentials, as described in (2.8), (2.9) and (2.10).
It is also invariant with respect to swapping the singularity at ∞ with finite singularities.
Thus assume that σ(z) = zρ(z). Then substitution w = z−1 transforms (3.19) into
σ̃(w)∂2w +
(
τ̃(w)− σ̃(w)
w − λ−1
)
∂w + η̃(w)− η̃
(
λ−1
)
− σ̃
(
λ−1
)
µ2
−
(
τ̃
(
λ−1
)
σ̃′
(
λ−1
))
µ+
σ̃
(
λ−1
)
µ
w − λ−1
,
where
σ̃(w) := w3ρ
(
w−1
)
, (3.20a)
τ̃(w) := w2
(
3ρ
(
w−1
)
− τ
(
w−1
))
, (3.20b)
η̃(w) := η
(
w−1
)
, (3.20c)
µ̃ := −λ2µ. (3.20d)
22 J. Dereziński, A. Ishkhanyan and A. Latosiński
Transformation w = z−1 transforms a deformed Heun class operator satisfying σ(0) = 0 into
another deformed Heun class operator satisfying σ̃(0) = 0, similarly as for undeformed Heun
class operators. Note however a subtle difference between (3.17b) and (3.20b).
The following proposition describes mapping properties of the above described transformation
in more detail:
Proposition 3.5.
1. ση(0) = 0⇔ deg σ̃η̃ ≤ 3.
2. σ′(0) = 0⇔ deg σ̃ ≤ 2.
3. σ′(0) = 0 and τ(0) = 0⇔ deg σ̃ ≤ 2 and deg τ̃ ≤ 1.
4. σ′(0) = 0 and (ση(0) = (ση)′(0) = 0⇔ deg σ̃ ≤ 2 and deg σ̃η̃ ≤ 2.
3.6 Classification of Heun class equations
In this subsection we discuss two classifications of Heun class equations and operators.
The first is based on the rank of singularities. We classify half-integer and integer ranks
separately, except for the rank 1, where we use, as usual, the rounded rank. Not counting the
types reducible to the Riemann class, which are treated as “trivial”, it partitions the Heun class
into ten types.
There exists also a coarser classification, which uses rounded singularity ranks. It groups the
ten nontrivial types of the Heun class into five supertypes.
In the following list we give both classifications of the Heun class:
� (standard) Heun or (1111).
� confluent Heun or (112).
– non-degenerate confluent Heun or (112).
– degenerate confluent Heun or
(
113
2
)
.
� doubly confluent Heun or (22).
– non-degenerate doubly confluent Heun or (22).
– degenerate doubly confluent Heun or
(
3
22
)
.
– doubly degenerate doubly confluent Heun or
(
3
2
3
2
)
.
� biconfluent Heun or
(
13
)
.
– non-degenerate biconfluent Heun or (13).
– degenerate biconfluent Heun
(
15
2
)
.
� triconfluent Heun (4).
– non-degenerate triconfluent Heun (4).
– degenerate triconfluent Heun
(
7
2
)
.
In the above list we use names similar to those proposed by [25].
Some of the types in this list have two distinct varieties, which are equivalent by swapping
a finite singularity with infinity. The variety where the higher rank singularity is put at ∞
is sometimes called the natural. For instance, (112) has the natural variety (11; 2) and the
alternative variety (21; 1).
For some varieties we give more than one normal form — they are labelled a) and b).
In the following theorem we describe normal forms of various types of Heun class operators.
Note that there is some arbitrariness in the choice of a normal form. We allow the following
transformations: z 7→ az+ b, division by a constant, sandwiching with powers and exponentials.
From Heun Class Equations to Painlevé Equations 23
Theorem 3.6. Each Heun class operator can be transformed into a Riemann class operator or
one of the following normal forms:
type σ(z) τ(z) η(z)
(111; 1) z(z − 1)(z − t) a2z
2 + a1z + a0
a) b1z + b0
b) b0 + b−1z
−1 t 6= 0, 1
(11; 2) z(z − 1) a2z
2 + a1z + a0
a) b1z + b0
b) b0 + b−1z
−1 a2 6= 0
(21; 1) z2(z − 1) a2z
2 + a1z + a0
a) b1z + b0
b) b0 + b−1z
−1 a0 6= 0(
11; 3
2
)
z(z − 1) a1z + a0 b1z + b0 b1 6= 0(
3
21; 1
)
z2(z − 1) a2z
2 + a1z b0 + b−1z
−1 b−1 6= 0
(2; 2) z2 a2z
2 + a1z + a0
a) b1z + b0
b) b0 + b−1z
−1 a2 6= 0, a0 = c(
3
2 ; 2
)
z2 a2z
2 + a1z b0 + b−1z
−1 b−1 6= 0, a2 = c(
2; 3
2
)
z2 a1z + a0 b1z + b0 b1 6= 0, a0 = c(
3
2 ; 3
2
)
z2 0 b1z + b0 + b−1z
−1 b−1 6= 0, b1 = c(
1; 3
2
)
z2 a1z b1z b1 = c(
3
2 ; 1
)
z2 a1z b−1z
−1 b−1 = c
(1; 3) z a2z
2 + a1z + a0
a) b1z + b0
b) b0 + b−1z
−1 a2 = c
(3; 1) z3 a2z
2 + a1z + a0
a) b1z + b0
b) b0 + b−1z
−1 a0 = c(
1; 5
2
)
z a0 b2z
2 + b1z + b0 b2 = c(
5
2 ; 1
)
z3 a2z
2 b0 + b−1z
−1 + b−2z
−2 b−2 = c
(; 4) 1 a2z
2 + a0 b1z + b0 a2 = c(
; 7
2
)
1 0 b3z
3 + b1z + b0 b3 = c
In the above table c denotes an arbitrary nonzero constant.
Proof. If σ has 3 distinct roots, it can be transformed to z(z−1)(z−t), t 6= 0, 1. By sandwiching
with powers at each finite singularity we can make one of indices 0. Then η becomes a polynomial
and deg η ≤ 1. We obtain the normal form of (111; 1).
Let σ have degree 2 and 2 distinct roots. It can be transformed to z(z − 1). At each finite
singularity we can make one of indices 0. Then η becomes a polynomial and deg η ≤ 2. By the
transformation e−κz · eκz with κ solving
κ2 + a2κ+ b2 = 0
we can make b2 = 0. If a2 6= 0 we obtain the normal form of (11; 2).
Assume that a2 = 0. If b1 = 0, we get the 2F1 operator, which belongs to the Riemann class.
Otherwise we obtain the normal form of
(
11; 3
2
)
.
Let σ have degree 2 and one root. It can be transformed to z2. We have
η(z) = b2z
2 + b1z + b0 + b−1z
−1 + b−2z
−2.
By e−κz · eκz with κ solving
κ2 + a2κ+ b2 = 0
we can kill b2. By eκz
−1 · e−κz−1
with κ solving
κ2 + a0κ+ b−2 = 0
we can kill b−2.
24 J. Dereziński, A. Ishkhanyan and A. Latosiński
Let a0 6= 0. By scaling we can make a0 = 1. Then by z−λ · zλ with λ = −b−1 we can kill b−1,
keeping a0 = 1. If a2 6= 0, we obtain the normal form of (2; 2). If a2 = 0 and b1 = 0, we
obtain 2F0 or Euler II type, both of the Riemann class. If a2 = 0 and b1 6= 0, we obtain the
normal form of
(
2; 3
2
)
.
Let a0 = 0. If a2 6= 0, by scaling we can make a2 = 1 Then by z−λ · zλ with λ = −b1 we can
kill b1 keeping a2 = 1. We obtain the normal form of
(
3
2 ; 2
)
.
Let a0 = a2 = 0. If b1 = b−1 = 0, the operator is of the Riemann class. If b−1 = 0, b1 6= 0, then
with z−λ ·zλ we kill b0 and we obtain z
(
z∂2z +a1∂z+b1
)
. The operator in brackets can be reduced
to the F1 operator. If b1 = 0, b−1 6= 0, we similarly kill b0 obtaining 1
z
(
z3∂2z + a1z
2∂z + b−1
)
.
The operator in brackets, after the transformation z 7→ 1
z can be transformed to a F1 operator.
If b−1, b1 6= 0 we apply z−λ · zλ with λ = −a1
2 to kill a1. We obtain the normal form of
(
3
2 ; 3
2
)
.
Let σ have degree 1. It can be transformed to z. One of indices at 0 can be made 0. Then η
becomes a polynomial of degree ≤ 3. By applying e−κz
2 · eκz2 with κ =
√
−b3
2 we can kill b3.
If a2 6= 0, applying e−κz · eκz with κ = − b2
a2
we kill b2. By scaling we can make a2 = 1 and we
obtain the normal form of (1; 3). If a2 = 0 and b2 6= 0, by applying e−κz · eκz with κ = −a1
we kill a1. If b2 6= 0, by scaling we can make b2 = 1 and we obtain the normal form of
(
1; 5
2
)
.
If b2 = 0, by applying e−κz · eκz with κ solving
κ2 + a1κ+ b1 = 0
we obtain an operator which can degenerate to the 1F1, 0F1 or Euler I type, all of the Riemann
class.
Let σ have degree 0. We can assume that it is 1. η is a polynomial of degree 4.
By applying e−κz
3 · eκz3 with 9κ2 + a23κ+ b4 = 0 we kill b4.
Let a2 6= 0. By applying e−κz
2 · eκz2 with κ = − b3
2a2
we kill b3. By applying e−κz · eκz with
κ = − b2
a2
we kill b2. After a transformation z 7→ az + b we can assume that τ(z) = z2 + a0.
We obtain the normal form of (; 4)
Let a2 = 0. By applying e−κz
2 · eκz2 with κ = −a1
4 we kill a1. By applying e−κz · eκz with
κ = −a0
2 we kill a0. Thus τ = 0. If b3 6= 0, then after a transformation z 7→ az + b we can
assume that τ(z) = z3 + a1z + a0. We obtain the normal form of
(
; 7
2
)
. If b3 = 0, we obtain an
operator that can degenerate to the Hermite, Airy, 1d Helmholtz or 1d Laplace type, all of the
Riemann class.
In (11; 2), (1; 3) and (2; 2) the transformation zλ · z−λ with λ = − b1
a2
kills b1z and produ-
ces b−1z
−1. Thus it makes the normal form b) out of the normal form a).
If σ has degree 3 and 2 distinct roots, it can be transformed to z2(z−1). The transformation
z 7→ z−1 leads to σ(z) = z(z − 1).
If σ has degree 3 and only 1 root, it can be transformed to z3. Then z 7→ z−1 yields
σ(z) = z. �
Remark 3.7. The operators listed in the table of Theorem 3.6 as
(
1; 3
2
)
and
(
3
2 ; 1
)
are strictly
speaking not of the Riemann class: they are z times an operators of the Riemann class. Hence
they yield equations of the Riemann class. So they can be considered as “trivial” and were
ignored in the table at the beginning of the subsection.
4 From Heun class to Painlevé equations
4.1 Method of isomonodromic deformations
Let us review the theory of isomonodromic deformations of linear second order differential equa-
tions following [12, 19, 20]. We shall use a notation similar to [19].
From Heun Class Equations to Painlevé Equations 25
Let p, q be rational functions of z, depending on some parameters. Among these parameters
we single out a parameter t. We will write v′ for ∂
∂zv and v̇ for ∂
∂tv. Consider a family of linear
second order differential equations of the form
v′′(z) + p(z)v′(z) + q(z)v(z) = 0. (4.1)
We assume that when we deform the equation (4.1), we can also deform its certain solution v
so that the following condition is satisfied:
v̇ = a(z, t)v′ + b(z, t)v. (4.2)
This essentially means that when we deform the equation, its solutions “live” on the same
Riemann surface. In particular, if there are singularities, then one should expect that the
monodromy of solutions stays the same.
The compatibility of (4.1) and (4.2) imposes a strong condition on the deformation. Indeed,
differentiating (4.2) in z we obtain
v̇′ = (a′ − ap+ b)v′ + (−aq + b′)v.
Differentiating (4.1) once in t and (4.2) twice in z we get
v̇′′ =
(
−ṗ− pa′ + ap2 − pb− qa
)
v′ +
(
paq − pb′ − q̇ − qb)v, (4.3)
v̇′′ =
(
−2qa′ − aq′ + b′′ + apq − bq
)
v +
(
a′′ − 2pa′ − ap′ + 2b′ − aq + ap2 − bp)v′. (4.4)
Equating (4.4) and (4.3) we obtain
ṗ− ap′ + 2b′ − pa′ + a′′ = 0, (4.5)
q̇ + pb′ − 2qa′ − aq′ + b′′ = 0. (4.6)
When applying this method to a concrete family of equations one needs to divide its param-
eters into two categories. The first category should contain all parameters responsible for the
monodromy around singular points. For example, the coefficients p−1 and q−2 of the Laurent
series of p, resp. q around singular points. In the second category we have parameters that
do not influence the monodromy, typically denoted µ, λ, t. The variable t is called the “time
variable”.
4.2 Isomonodromic deformations in presence of a non-logarithmic singularity
Let σ, τ , η be rational functions. (At the moment we do not assume the conditions (3.15) for
the Heun class). Consider the differential equation given by
∂2z + p0(z)∂z + q0(z) = ∂2z +
τ(z)
σ(z)
∂z +
η(z)
σ(z)
. (4.7)
We assume that σ, τ , η depend on a parameter t. Let λ, µ be additional parameters. Following
the prescription of (2.28), we introduce the deformed equation corresponding to (4.7):
∂2z + p(λ, z)∂z + q(λ, µ, z) = ∂2z +
(
τ(z)
σ(z)
− 1
z − λ
)
∂z
+
1
σ(z)
(
η(z)− η(λ)− µ
(
τ(λ)− σ′(λ)
)
− µ2σ(λ) +
µσ(λ)
(z − λ)
)
. (4.8)
The following theorem is devoted to monodromic deformations of (4.8). It unifies a large fam-
ily of cases in a single formulation. Unfortunately, this unification has one drawback: relatively
complicated conditions (4.9), (4.10) and (4.11) constraining the choice of the time variable t and
the auxiliary polynomial c. Probably this drawback is impossible to avoid. The main results of
our paper, described in the next subsection, will be corollaries of Theorem 4.1.
26 J. Dereziński, A. Ishkhanyan and A. Latosiński
Theorem 4.1. Suppose that c(z) = c(t, λ, z) is a t, λ-dependent polynomial of degree ≤ 2.
Suppose that the following conditions are satisfied:
0 =
τ̇
σ
(z)− τ σ̇
σ2
(z) +
cτ
σ (z)− cτ
σ (λ)− (z − λ)
(
cτ
σ
)′
(z)
(z − λ)2
, (4.9)
0 = − σ̇
σ
(z)
(
η(z)− η(λ)
)
+ η̇(z)− η̇(λ)
+
η(z)− η(λ)
(z − λ)
(
cσ′
σ
(z)− c′(z)
)
− η′(λ)
(
σ′c
σ
(λ)− c′(λ)
)
+
2cη(z)− 2cη(λ)−
(
(cη)′(z) + (cη)′(λ)
)
(z − λ)
(z − λ)2
, (4.10)
0 =
σ̇
σ
(z)− σ̇
σ
(λ)−
cσ′
σ (z)− cσ′
σ (λ)−
(
cσ′
σ
)′
(λ)(z − λ)
(z − λ)
. (4.11)
Define the compatibility functions
a(t, λ, z) :=
c(z)
z − λ
, b(t, λ, µ, z) = −c(λ)µ
z − λ
. (4.12)
Then the following equations for λ, µ
λ̇ = 2c(λ)µ− c′(λ) +
cτ
σ
(λ), (4.13)
µ̇ = −cη
′
σ
(λ)− µ
(
cτ ′
σ
(λ)− cσ′′
2σ
(λ)− c′′
2
)
− µ2σ
′c
σ
(λ) (4.14)
are equivalent to the compatibility conditions (4.6) and (4.5).
The proof of Theorem 4.1 is deferred to Appendixes A.1, A.2 and A.3.
4.3 Isomonodromic deformations of Heun class equations
This subsection contains the main results of our paper. We will suppose that σ is a polynomial
of degree ≤ 3, τ is a polynomial of degree ≤ 2 and ησ is a polynomial of degree ≤ 4. In other
words, we will assume that (4.7) is a Heun class equation. We will show that Theorem 4.1 can
be applied to a large family of Heun class equations, including normal forms of all its types.
As a result we obtain all types of Painlevé equation.
Our main results will be formulated in two theorems. In Theorem 4.2 we still try to give
a unified treatment. More precisely, we consider two closely related ansatzes, which we call A
and B. Ansatz A is applicable if σ has a zero. Ansatz B can be used if the degree of σ is ≤ 2.
The time variable is not specified, it is only constrained by certain conditions.
In Theorem 4.3 the time variable is always explicit. Unfortunately, it seems impossible to do
it in a unified way — we are compelled to consider 5 distinct cases. (Note that 5 is still less
than the number of types of Heun class equations. Besides, some of these cases are applicable
to more than one type).
Theorem 4.2. Case A. Assume that s ∈ C and σ(s) = 0, so that we can write σ(z) = (z−s)ρ(z)
for a polynomial ρ of degree ≤ 2. We assume that σ, τ , η depend on t. Let m be a function of t
satisfying the following conditions
∂t
τ
σ
(z) =
mτ(s)
(z − s)2
, (4.15)
From Heun Class Equations to Painlevé Equations 27
σ̇
σ
(z)
(
η(z)− η(λ)
)
− η̇(z) + η̇(λ) = m
((
η(z)− η(λ)
)
(λ− s)ρ(z)
(z − λ)(z − s)
− η′(λ)ρ(λ)
)
+
(λ− s)
(z − λ)2
(
2ρη(z)− 2ρη(λ)−
(
(ρη)′(z) + (ρη)′(λ)
)
(z − λ)
)
, (4.16)
σ̇
σ
(z)− σ̇
σ
(λ) = mρ(s)
(z − λ)
(z − s)(λ− s)
. (4.17)
Define the compatibility functions
a(t, λ, z) =
m(λ− s)ρ(z)
z − λ
, b(t, λ, z) = −m(λ− s)ρ(λ)µ
z − λ
,
and the Hamiltonian
H(t, λ, µ) = m
(
η(λ) +
(
τ(λ)− (λ− s)ρ′(λ)
)
µ+ σ(λ)µ2
)
. (4.18)
Then λ, µ satisfy the Hamilton equations with respect to H, that is
dλ
dt
=
∂H
∂µ
(t, λ, µ),
dµ
dt
= −∂H
∂λ
(t, λ, µ),
if and only if (4.6) and (4.5) hold.
Case B. Assume that deg σ ≤ 2 and deg ση ≤ 3. Suppose that τ , η, but not σ depend on t.
Let m be a function of t satisfying
τ̇(z)
σ(z)
= m
τ ′′
2
, (4.19)
η̇(z)− η̇(λ) = m
(ση)′′′
6
(z − λ). (4.20)
Define the compatibility functions
a(t, λ, z) =
mσ(z)
z − λ
, b(t, λ, z) = −mσ(λ)µ
z − λ
,
and the Hamiltonian
H(t, λ, µ) := m
(
η(λ) +
(
τ(λ)− σ′(λ)
)
µ+ σ(λ)µ2
)
. (4.21)
Then λ, µ satisfy the Hamilton equations with respect to H if and only if (4.6) and (4.5) hold.
Proof. We set
c(z) = m(λ− s)ρ(z), in Case A, (4.22)
c(z) = mσ(z), in Case B. (4.23)
We apply Theorem 4.1. More precisely, we check that conditions (4.9), (4.10) and (4.11) are
equivalent to (4.15), (4.16) and (4.17), resp. to (4.19), (4.20). We also verify that equations (4.13)
and (4.14) coincide with the Hamilton equations for H(t, λ, µ).
Details of computations are given in Appendixes A.4, A.5 and A.6. �
28 J. Dereziński, A. Ishkhanyan and A. Latosiński
Note that it is possible to unify the formulas (4.18) and (4.21) for the Hamiltonian in a single
formula using the polynomial c from (4.22) and (4.23):
H(t, λ, µ) =
η(λ)c(λ)
σ(λ)
+ µ
(
τ(λ)c(λ)
σ(λ)
− c′(λ)
)
+ µ2c(λ).
Ansatzes A and B of Theorem 4.2 will still be subdivided into several subcases that differ by
the choice of the time variable. All these subcases are described in Theorem 4.3 below, which
together with Theorem 4.2 describes the main result of our paper.
The main subcase of Ansatz A is A1, where the time variable is the position of a nondegenerate
zero of σ. This is of course generically true, however there are situations when σ does not have
nondegenerate zeros. Subcases Ap and Aq can be applied when σ has a degenerate (that is, at
least double) zero.
In Subcases Ap and Bp time is contained in the function p0 (the coefficient of the first order
term) of (4.7) and in Aq and Bq it is contained in q0. Subcases Ap and Bp are typically used
when the rank at ∞ is an integer. Subcases Aq and Bq are more appropriate for degenerate
types, when the rank at ∞ is half-integer.
In the following theorem, first we describe σ, τ and η that belong to a given subcase, specifying
explicitly the dependence on t. Next we write the corresponding (undeformed) Heun class
operator in the principal form. Then we give the corresponding compatibility functions a, b and
the Hamiltonian H.
By writing deg β ≤ n we mean that β is a polynomial in z of degree ≤ n. Unlike in Theo-
rem 4.3, the dependence on the parameter t will be always explicitly given.
Theorem 4.3.
� Subcase A1. Let
σ(z) = (z − t)ρ(z), deg ρ ≤ 2;
τ(z) = (1− κ)ρ(z) + φ(z)(z − t), κ ∈ C, deg φ ≤ 1;
η(z) =
αρ(t)
z − t
+ η0(z) α ∈ C, deg η0 ≤ 1.
Consider
∂2z +
(1− κ
z − t
+
φ(z)
ρ(z)
)
∂z +
αρ(t)
(z − t)2ρ(z)
+
η0(z)
(z − t)ρ(z)
.
If ρ(t) 6= 0, then Theorem 4.2A holds with m = ρ(t)−1,
a(z) :=
(λ− t)ρ(z)
ρ(t)(z − λ)
, b(z) := − σ(λ)µ
ρ(t)(z − λ)
,
ρ(t)H := (λ− t)ρ(λ)µ2 +
(
(1− κ)ρ(λ) + (λ− t)(φ(λ)− ρ′(λ)
)
µ+
αρ(t)
λ− t
+ η0(λ).
� Subcase Ap. Let
σ(z) = (z − s)2ρ1(z), s ∈ C, deg ρ1 ≤ 1;
τ(z) = tρ1(z) + τ0(z), deg τ0 ≤ 2;
η(z) =
ψ(z)
ρ1(z)
, degψ ≤ 2.
From Heun Class Equations to Painlevé Equations 29
Consider
∂2z +
(
t
(z − s)2
+
τ0(z)
(z − s)2ρ1(z)
)
∂z +
ψ(z)
(z − s)2ρ1(z)2
.
If τ0(s) 6= 0 or ρ1(s) 6= 0, then Theorem 4.2A holds with m =
(
τ0(s) + tρ1(s)
)−1
,
a(z) :=
(λ− s)(z − s)ρ1(z)(
τ0(s) + tρ1(s)
)
(z − λ)
, b(z) := − (λ− s)2ρ1(λ)µ(
τ0(s) + tρ1(s)
)
(z − λ)
,(
τ0(s) + tρ1(s)
)
H := (λ− s)2ρ1(λ)µ2
+
(
tρ1(λ) + τ0(λ)− (λ− s)ρ1(λ)− (λ− s)2ρ′1
)
µ+
ψ(λ)
ρ1(λ)
.
� Subcase Aq. Let
σ(z) := (z − s)2ρ1(z), s ∈ C, deg ρ1 ≤ 1;
τ(z) = (z − s)φ(z), deg φ ≤ 1;
η(z) =
t
z − s
+ η0(z), deg ρ1η0 ≤ 2.
Consider
∂2z +
φ(z)
(z − s)ρ1(z)
∂z +
1
(z − s)2ρ1(z)
(
t
z − s
+ η0(z)
)
.
If (ση0)
′(s) 6= 0 or ρ1(s) 6= 0, then Theorem 4.2A holds with m =
(
(ση0)
′(s) + ρ1(s)t
)−1
:
a(z) :=
(λ− s)(z − s)ρ1(z)(
(ση0)′(s) + ρ1(s)t
)
(z − λ)
, b(z) := − (λ− s)2ρ1(λ)µ(
(ση0)′(s) + ρ1(s)t
)
(z − λ)
,(
(ση0)
′(s) + ρ1(s)t
)
H := (λ− s)2ρ1(λ)µ2 +
(
(λ− s)(φ(λ)− ρ1(λ))− (λ− s)2ρ′1
)
µ
+
t
λ− s
+ η0(λ).
� Subcase Bp. Let
σ(z), deg σ ≤ 2;
τ(z) = tσ(z) + τ0(z), deg τ0 ≤ 2;
η(z), deg ση ≤ 2.
Consider
∂2z +
(
t+
τ0(z)
σ(z)
)
∂z +
η(z)
σ(z)
.
If σ′′ 6= 0 or τ ′′0 6= 0, then Theorem 4.2B holds with m =
(
tσ
′′
2 +
τ ′′0
2
)−1
:
a(z) :=
σ(z)(
tσ
′′
2 +
τ ′′0
2
)
(z − λ)
, b(z) := − σ(λ)µ(
tσ
′′
2 +
τ ′′0
2
)
(z − λ)
,(
t
σ′′
2
+
τ ′′0
2
)
H := σ(λ)µ2 +
(
tσ(λ) + τ0(λ)− σ′(λ)
)
µ+ η(λ).
30 J. Dereziński, A. Ishkhanyan and A. Latosiński
� Subcase Bq. Let
σ(z), deg σ ≤ 2;
τ(z), deg τ ≤ 1;
η(z) = tz + η0(z), deg ση0 ≤ 3.
Consider
∂2z +
τ(z)
σ(z)
∂z +
tz
σ(z)
+
η0(z)
σ(z)
.
If σ′′ 6= 0 or (ση0)
′′′ 6= 0, then Theorem 4.2B holds with m =
(
tσ
′′
2 + (ση0)′′′
6
)−1
,
a(z) :=
σ(z)(
tσ
′′
2 + (ση0)′′′
6
)
(z − λ)
, b(z) := − σ(λ)µ(
tσ
′′
2 + (ση0)′′′
6
)
(z − λ)
,(
t
σ′′
2
+
(ση0)
′′′
6
)
H := σ(λ)µ2 +
(
τ(λ)− σ′(λ)
)
µ+ tλ+ η0(λ).
Note that one can deduce Subcases Ap and Aq from Subcases Bp resp. Bq by applying the
symmetry described above Proposition 3.5. However, in Appendices A.4, A.5 and A.6, where
we prove Theorem 4.3, we will give independent proofs of all subcases.
Note also that the union of subcases of Theorem 4.3 does not cover the whole Heun class.
However, it covers all appropriately interpreted normal forms listed in Theorem 3.6. This will
be further discussed in the following subsection.
Remark 4.4. In our applications we will sometimes use rescaled versions of the above construc-
tions. In fact, if ε 6= 0, we replace t with εt and multiply a, b, H with ε, then the above theorem
remains true.
4.4 Correspondence between Heun class and Painlevé equations
Traditionally, Painlevé equations are divided into 6 types: Painlevé I–VI. However, one can
argue that some of their degenerate cases should be treated as separate types.
Thus Painlevé V (5.1) splits into the nondegenerate Painlevé V with δ 6= 0 and the degenerate
Painlevé V with δ = 0. We denote the former simply by ndeg-V and the latter by deg-V. One
can show that deg-V Painlevé is equivalent to Painlevé III′, however it is natural to treat it as
a separate type. All that is explained in Section 5.2.
With Painlevé III (5.5) the situation is more complicated. First of all, following various
authors, we prefer to use the Painlevé III′ equation, which is equivalent to Painlevé III by a simple
transformation. Beside the nondegenerate case we have the degenerate case and the doubly
degenerate case. We denote them respectively, ndeg-III′, deg-III′ and ddeg-III′.
(
Ohyama–Oku-
mura denote them
(
D
(1)
6
)
,
(
D
(1)
7
)
,
(
D
(1)
8
)
.
)
One can also consider an alternative degenerate case
γ 6= 0, δ = 0, which is however equivalent to deg-III′.
(
Ohyama–Okumura denotes it
(
D
(1)
7
)
−2.
)
All of that is explained in Section 5.3.
Finally, it is natural to consider the Painlevé 34 equation (4.40), which can be viewed as a
degenerate case of the Painlevé IV equation (5.10). One can show that Painlevé 34 is equivalent
to Painlevé II, however it is natural to keep it as a separate type. See Section 5.4 for more
comments.
This way we obtain 10 types of Painlevé equations. Recall that we also have 10 types of Heun
class equations. In fact, each type of Painlevé can be derived from one of the types of deformed
Heun. Here is the list of correspondences:
From Heun Class Equations to Painlevé Equations 31
(standard) Heun (111; 1) A1 a),b) Painlevé VI (1111)
ndeg. confluent Heun
(11; 2) A1 a),b), Bp b)
Painlevé ndeg-V (112)
(21; 1) A1 a), Ap a)
deg. confluent Heun
(
11; 3
2
)
A1, Bq
Painlevé deg-V
(
11 3
2
)(
3
21; 1
)
Aq
db. confluent Heun (2; 2) Ap a), Bp b) Painlevé ndeg-III′ (22)
deg. db. confluent Heun
(
3
2 ; 2
)
Aq, Bp
Painlevé deg-III′
(
3
22
)(
2; 3
2
)
Ap, Bq
ddeg. db. confluent Heun
(
3
2 ; 3
2
)
Bq Painlevé ddeg-III′
(
3
2
3
2
)
ndeg. bi-confluent Heun
(1; 3) A1 a),b), Bp a),b)
Painlevé IV (13)
(3; 1) Ap b)
deg. bi-confluent Heun
(
1; 5
2
)
Bq
Painlevé 34
(
1 5
2
)
( 5
2 ; 1) Aq
ndeg. tri-confluent Heun (; 4) Bp Painlevé II (4)
deg. tri-confluent Heun
(
; 7
2
)
Bq Painlevé I
(
7
2
)
In the first column we give the name of the Heun class type. For typographical reasons we
abbreviate “nondegenerate” to ndeg, “degenerate” to deg, “doubly degenerate” to ddeg and
“doubly” to db.
In the second column we give the symbol of the type in terms of the ranks of singularities.
We also indicate which singularity is at∞. In several cases there are two possibilities — we give
both of them.
In the third column we indicate subcases of Theorem 4.3 which can be applied to certain
normal forms of a given type. Normal forms are taken from the table in Theorem 3.6. If in that
table more than one normal form is given, a) or b) indicates which normal form is considered.
(In some cases, the normal forms from Theorem 3.6 need to be slightly modified: When we
apply A1 to the form b) of (111; 1) we change the roles of the roots; for (11; 2) and (1; 3) we
shift one root from 0 to t; finally for (21; 1) and
(
11; 3
2
)
we shift a root from 1 to t.)
In the fourth column we give the name of the Painlevé type that can be obtained by the
isomonodromic deformation.
In the fifth column we list the symbol in terms of ranks of singularities without the indication
of the position of ∞. We will often use it in the sequel as the name of the given type of
the Painlevé equation. Thus, e.g., the Painlevé
(
3
22
)
equation is an alternative name for the
degenerate Painlevé III′ equation. We will actually prefer these names to the traditional ones,
similarly as Slavyanov–Lay in [25].
Occasionally, we will also use the names for Painlevé equation involving the position of∞. For
instance, the Painlevé
(
3
2 ; 2
)
equation will mean the form of the degenerate Painlevé III′ equation
obtained from the Heun
(
3
2 ; 2
)
equation. The Painlevé
(
2; 3
2
)
equation will denote the equation
obtained from the Heun
(
2; 3
2
)
equation. Both forms of Painlevé equation are equivalent.
We will discuss further the classification of Painlevé equations in Section 5, where we will see
how to group the 10 types into 5 supertypes, parallel to the grouping of 10 types of Heun class
equations into 5 supertypes.
In the following subsections we describe how to obtain all types of Painlevé equations from
deformed Heun class equations. First we give the functions σ, τ , η describing one of possible
normal forms of a given type of the Heun class equation. We indicate explicitly the dependence
of σ, τ , η on the time variable t. Then we present this equation in its principal form. Next we
give the corresponding deformed equation. Next we give the compatibility functions a, b and
the Painlevé Hamiltonian. Finally, we describe the resulting Painlevé equation.
32 J. Dereziński, A. Ishkhanyan and A. Latosiński
Note that the whole procedure is determined by σ, τ , η, by the choice of the time variable t
and the functions a, b. The latter are restricted by Theorem 4.3. We always indicate which case
of Theorem 4.3 we use.
In our derivations we follow the paper of Ohyama–Okumura [19]. We have slightly changed
their notation for some of the parameters. We parametrize the equations by the differences of
indices at singular points of the deformed equation. In particular, if the rounded rank at z0
is 1, the parameter is called κz0 , if the rank is 2, it is called χz0 and if the rank is 3 it is
called θz0 .
One of the parameters of the initial Heun class equation — the free term in η — does not
enter in the deformed equation, and therefore is not used by [19]. We denote it simply by c.
Note that there is some arbitrariness in the choice of Hamiltonians, where a term depending
on t, but not on λ, µ, can always be added. We always choose Hamiltonians coinciding with
those of [19].
If in a given type σ has a root of multiplicity 1, one can usually use Case A1 of Theorem 4.3.
Following [19], we use it only for Heun (1111)–Painlevé VI. For the other types we use Ap, Aq,
Bp, Bq.
In general, for each type of Heun class we give one derivation. The exception is the type(
23
2
)
, where, following again [19], we give two versions of derivations of deg-III′: one in the form
of
(
2; 3
2
)
, the other in the form of
(
3
2 ; 2
)
.
4.5 From Heun (111;1) to Painlevé VI
Set
σ(z) = z(z − 1)(z − t),
τ(z) = (1− κ0)(z − 1)(z − t) + (1− κ1)z(z − t) + (1− κt)z(z − 1),
η(z) =
(
(κ0 + κ1 + κt − 1)2 − κ2∞
)
z
4
− c.
Heun (111; 1) equation
∂2z+
(
1− κ0
z
+
1− κ1
z− 1
+
1− κt
z− t
)
∂z+
1
z(z− 1)(z− t)
((
(κ0+ κ1+ κt− 1)2− κ2∞
)
4
z− c
)
.
Deformed Heun (111; 1) equation
∂2z +
(
1− κ0
z
+
1− κ1
z − 1
+
1− κt
z − t
− 1
z − λ
)
∂z
+
1
z(z − 1)(z − t)
((
(κ0 + κ1 + κt − 1)2 − κ2∞
)
4
(z − λ)− µ2λ(λ− 1)(λ− t)
+
(
κ0(λ− 1)(λ− t) + κ1λ(λ− t) + κtλ(λ− 1)
)
µ+
λ(λ− 1)(λ− t)µ
(z − λ)
)
.
Type A1 compatibility functions
a(z) =
(λ− t)z(z − 1)
t(t− 1)(z − λ)
, b(z) = −λ(λ− 1)(λ− t)µ
t(t− 1)(z − λ)
.
Painlevé (111; 1) Hamiltonian
t(t− 1)H = λ(λ− 1)(λ− t)µ2 −
(
κ0(λ− 1)(λ− t) + κ1λ(λ− t) + (κt − 1)λ(λ− 1)
)
µ
+
(
(κ0 + κ1 + κt − 1)2 − κ2∞
)
(λ− t)
4
.
From Heun Class Equations to Painlevé Equations 33
Painlevé (111; 1) equation
d2λ
dt2
=
1
2
(
1
λ
+
1
λ− 1
+
1
λ− t
)(
dλ
dt
)2
−
(
1
t
+
1
t− 1
+
1
λ− t
)
dλ
dt
+
λ(λ− 1)(λ− t)
t2(t− 1)2
(
α+ β
t
λ2
+ γ
t− 1
(λ− 1)2
+ δ
t(t− 1)
(λ− t)2
)
, (4.24)
where
α =
1
2
κ2∞, β = −1
2
κ20, γ =
1
2
κ21, δ =
1
2
(
1− κ2t
)
.
The standard name of (4.24) is Painlevé VI equation.
4.6 From Heun (21;1) to Painlevé V
Set
σ(z) = (z − 1)2z,
τ(z) = (2− χ1)z(z − 1) + (1− κ0)(z − 1)2 + tz,
η(z) =
(
(κ0 + χ1 − 1)2 − κ2∞
)
(z − 1)
4
− c.
Heun (21; 1) equation (with the singularity of rank 2 put at 1):
∂2z +
(
2− χ1
z − 1
+
1− κ0
z
+
t
(z − 1)2
)
∂z +
1
(z − 1)2z
(
(κ0 + χ1 − 1)2 − κ2∞
4
(z − 1)− c
)
.
Deformed Heun (21; 1) equation:
∂2z +
(
t
(z − 1)2
+
2− χ1
z − 1
+
1− κ0
z
− 1
z − λ
)
∂z
+
1
(z − 1)2z
(
(κ0 + χ1 − 1)2 − κ2∞
4
(z − λ)− (λ− 1)2λµ2
−
(
−κ0(λ− 1)2 − χ1λ(λ− 1) + tλ
)
µ+
(λ− 1)2λµ
(z − λ)
)
.
Type Ap compatibility functions
a(z) =
(λ− 1)z(z − 1)
t(z − λ)
, b(z) = −(λ− 1)2λµ
t(z − λ)
.
Painlevé (21; 1) Hamiltonian
tH = (λ− 1)2λµ2 −
(
κ0(λ− 1)2 + (χ1 − 1)λ(λ− 1)− tλ
)
µ
+
(
(κ0 + χ1 − 1)2 − κ2∞
)
(λ− 1)
4
. (4.25)
Painlevé (21; 1) equation:
d2λ
dt2
=
(
1
2λ
+
1
λ− 1
)(
dλ
dt
)2
− 1
t
dλ
dt
+
(λ− 1)2
t2
(
αλ+
β
λ
)
+ γ
λ
t
− λ(λ+ 1)
2(λ− 1)
, (4.26)
where
α =
1
2
κ2∞, β = −1
2
κ20, γ = χ1.
According to the standard terminology, (4.26) is the nondegenerate case of the Painlevé V
equation.
34 J. Dereziński, A. Ishkhanyan and A. Latosiński
4.7 From Heun
(
3
2
1; 1
)
to degenerate Painlevé V
Set
σ(z) = (z − 1)2z,
τ(z) = (z − 1)z + (1− κ0)(z − 1)2,
η(z) = − t
(z − 1)
+
(
κ20 − κ2∞
)
z
4
− c.
Heun
(
3
21; 1
)
equation (with the singularity of rank 2 put at 1):
∂2z +
(
1
z − 1
+
1− κ0
z
)
∂z +
1
z(z − 1)2
(
− t
(z − 1)
+
(κ20 − κ2∞)
4
(z − 1)− c
)
.
Deformed Heun
(
3
21; 1
)
equation:
∂2z +
(
1
z− 1
+
1− κ0
z
− 1
z− λ
)
∂z +
1
(z − 1)2z
(
− t
(z − 1)
+
t
(λ− 1)
+
(κ20 − κ2∞)
4
(z − λ)
−λ(λ− 1)2µ2 +
(
λ(λ− 1) + κ0(λ− 1)2
)
µ+
(λ− 1)2λµ
(z − λ)
)
.
Type Aq compatibility functions:
a(z) =
(λ− 1)z(z − 1)
t(z − λ)
, b(z) = −λ(λ− 1)2µ
t(z − λ)
.
Painlevé
(
3
21; 1
)
Hamiltonian:
tH = λ(λ− 1)2µ2 − κ0(λ− 1)2µ+
(κ20 − κ2∞)(λ− 1)
4
− tλ
(λ− 1)
. (4.27)
Painlevé (321; 1) equation:
d2λ
dt2
=
(
1
2λ
+
1
λ− 1
)(
dλ
dt
)2
− 1
t
dλ
dt
+
(λ− 1)2
t2
(
αλ+
β
λ
)
− 2
λ
t
, (4.28)
where
α =
1
2
κ2∞, β = −1
2
κ20.
According to the standard terminology (4.28) is the degenerate Painlevé V equation.
4.8 From Heun (2;2) to non-degenerate Painlevé III′
Set
σ(z) = z2, τ(z) = t+ (2− χ0)z − z2, η(z) =
(χ0 + χ∞ − 1)z
2
− c.
Heun (2; 2) equation:
∂2z +
(
t
z2
+
2− χ0
z
− 1
)
∂z +
(χ0 + χ∞ − 1)
2z
− c
z2
.
From Heun Class Equations to Painlevé Equations 35
Deformed Heun (2; 2) equation:
∂2z +
(
t
z2
+
2− χ0
z
− 1− 1
z − λ
)
∂z
+
1
z2
(
(χ0 + χ∞ − 1)
2
(z − λ)− λ2µ2 −
(
t− χ0λ− λ2
)
µ+
λ2µ
(z − λ)
)
.
Type Ap compatibility functions:
a(z) :=
λz
t(z − λ)
, b(z) = − λ2µ
t(z − λ)
.
Painlevé (2; 2) Hamiltonian:
tH := λ2µ2 −
(
λ2 + (χ0 − 1)λ− t
)
µ+
1
2
(χ0 + χ∞ − 1)λ. (4.29)
Painlevé (2; 2) equation:
d2λ
dt2
=
1
λ
(
dλ
dt
)2
− 1
t
dλ
dt
+
αλ2
4t2
+
λ3
t2
+
β
4t
− 1
λ
, (4.30)
where
α = −4χ∞, β = 4χ0.
According to the standard terminology (4.30) could be called the nondegenerate Painlevé III′
equation.
4.9 From Heun
(
2; 3
2
)
to degenerate Painlevé III′
We set
σ(z) := z2, τ(z) = t+ z(2− χ0), η(z) =
z
2
− c.
Heun
(
2; 3
2
)
equation:
∂2z +
(
t
z2
+
2− χ0
z
)
∂z +
1
z2
(
1
2
z − c
)
.
Deformed Heun
(
2; 3
2
)
equation:
∂2z +
(
t
z2
+
2− χ0
z
− 1
z − λ
)
∂z +
1
z2
(
(z − λ)
2
− λ2µ2 −
(
t− χ0λ
)
µ+
λ2µ
(z − λ)
)
.
Type Ap compatibility functions
a(z) :=
λz
t(z − λ)
, b(z) = − λ2µ
t(z − λ)
.
Painlevé
(
2; 3
2
)
Hamiltonian:
tH = λ2µ2 +
(
1− χ0)λ+ t
)
µ+
λ
2
. (4.31)
Painlevé
(
2; 3
2
)
equation:
d2λ
dt2
=
1
λ
(
dλ
dt
)2
− 1
t
dλ
dt
− λ2
t2
+
β
4t
− 1
λ
, (4.32)
where
β = 4χ0.
According to the standard terminology (4.32) is one of the forms of the degenerate Painlevé III′
equation.
36 J. Dereziński, A. Ishkhanyan and A. Latosiński
4.10 From Heun
(
3
2
; 2
)
to degenerate Painlevé III′
Set
σ(z) = z2, τ(z) = −z2 + z, η(z) =
t
2z
− c+
χ∞z
2
.
Heun
(
3
2 ; 2
)
equation:
∂2z +
(
−1 +
1
z
)
∂z +
1
z2
(
t
2z
− c+
χ∞
2
z
)
.
Deformed Heun
(
3
2 ; 2
)
equation:
∂2z +
(
−1 +
1
z
− 1
z − λ
)
∂z +
1
z2
(
t
2z
− t
2λ
+
χ∞
2
(z − λ)− λ2µ2 +
(
λ2 + λ
)
µ+
µλ2
(z − λ)
)
.
Type Aq compatibility functions:
a(z) :=
λz
t(z − λ)
, b(z) = − λ2µ
t(z − λ)
.
Painlevé
(
3
2 ; 2
)
Hamiltonian
tH = λ2µ2 − λ2µ+
χ∞λ
2
+
t
2λ
. (4.33)
Painlevé
(
3
2 ; 2
)
equation:
d2λ
dt2
=
1
λ
(
dλ
dt
)2
− 1
t
dλ
dt
+
αλ2
4t2
+
λ3
t2
+
1
t
, (4.34)
where
α = −4χ∞.
According to the standard terminology (4.32) is one of the forms of the degenerate Painlevé III′
equation.
4.11 From Heun
(
3
2
; 3
2
)
to doubly degenerate Painlevé III′
We set
σ(z) = z2, τ(z) = 2z, η(z) =
z
2
− c+
t
2z
.
Heun
(
3
2 ; 3
2
)
equation
∂2z +
2
z
∂z +
1
z2
(1
2
z − c+
t
2z
)
.
Deformed Heun
(
3
2 ; 3
2
)
equation:
∂2z +
(
2
z
− 1
z − λ
)
∂z +
1
z2
(
t
2z
− t
2λ
+
1
2
(z − λ)− λ2µ2 +
µλ2
(z − λ)
)
.
From Heun Class Equations to Painlevé Equations 37
Type Aq compatibility functions:
a(z) :=
λz
t(z − λ)
, b(z) = − λ2µ
t(z − λ)
.
Painlevé
(
3
2 ; 3
2
)
Hamiltonian
tH = λ2µ2 + λµ+
λ
2
+
t
2λ
. (4.35)
Painlevé
(
3
2 ; 3
2
)
equation:
d2λ
dt2
=
1
λ
(
dλ
dt
)2
− 1
t
dλ
dt
− λ2
t2
+
1
t
. (4.36)
Following the standard terminology (4.36) could be called the doubly degenerate Painlevé III′
equation.
4.12 From Heun (1;3) to Painlevé IV
Set
σ(z) = z, τ(z) = 1− κ0 − tz −
z2
2
, η(z) =
θ∞z
2
− c.
Heun (1; 3) equation
∂2z +
(
1− κ0
z
− t− z
2
)
∂z +
1
z
(
θ∞
2
z − c
)
.
Deformed Heun (1; 3) equation
∂2z+
(
1− κ0
z
− t− z
2
− 1
z− λ
)
∂z+
1
z
(
θ∞(z− λ)
2
− λµ2+
(
1
2
λ2+ tλ+ κ0
)
µ+
λµ
(z− λ)
)
.
Type Bp compatibility functions:
a(z) :=
2z
(z − λ)
, b(z) = − 2λµ
(z − λ)
.
Painlevé (1; 3) Hamiltonian:
H = 2λµ2 − (λ2 + 2tλ+ 2κ0)µ+ θ∞λ. (4.37)
Painlevé (1; 3) equation
d2λ
dt2
=
1
2λ
(
dλ
dt
)2
+
3
2
λ3 + 4tλ2 + 2
(
t2 − α
)
λ+
β
λ
, (4.38)
where
α = −κ0 + 2θ∞ − 1, β = −2κ20.
In the standard terminology (4.38) is called the Painlevé IV equation.
38 J. Dereziński, A. Ishkhanyan and A. Latosiński
4.13 From Heun
(
1; 5
2
)
to Painlevé 34
Set
σ(z) = z, τ(z) = 1− κ0, η(z) = −1
2
z2 − 1
2
tz − c.
Heun
(
1; 5
2
)
equation:
∂2z +
1− κ0
z
∂z +
1
z
(
−1
2
z2 − tz
2
− c
)
.
Deformed Heun
(
1; 5
2
)
equation:
∂2z +
(
1− κ0
z
− 1
z − λ
)
+
1
z
(
−z
2
2
+
λ2
2
− t
2
(z − λ)− λµ2 + κ0µ+
λµ
z − λ
)
.
Type Bq compatibility functions:
a(z) =
z
z − λ
, b(z) = − λµ
z − λ
.
Painlevé
(
1; 5
2
)
Hamiltonian
H = λµ2 − κ0µ−
λ2
2
− tλ
2
. (4.39)
Painlevé
(
1; 5
2
)
equation:
d2λ
dt2
=
1
2λ
(dλ
dt
)2
+ 2λ2 + tλ− α
2λ
, (4.40)
where α = κ20.
According to [19], the standard name for (4.40) is the Painlevé 34 equation.
4.14 From Heun (;4) to Painlevé II
Set
σ(z) = 1, τ(z) = −2z2 − t, η(z) = −(2α+ 1)z − c.
Heun (; 4) equation:
∂2z −
(
2z2 + t
)
∂z − (2α+ 1)z − c.
Deformed Heun (; 4) equation:
∂2z +
(
−2z2 − t− 1
z − λ
)
∂z − (2α+ 1)(z − λ)− µ2 +
(
2λ2 + t
)
µ+
µ
z − λ
.
Type Bp compatibility functions scaled with ε = −1:
a(z) :=
1
2(z − λ)
, b(z) = − µ
2(z − λ)
.
Painlevé (; 4) Hamiltonian:
H =
1
2
µ2 −
(
λ2 +
t
2
)
µ−
(
α+
1
2
)
λ. (4.41)
Painlevé (; 4) equation:
d2λ
dt2
= 2λ3 + tλ+ α. (4.42)
According to the standard terminology (4.42) is called the Painlevé II equation
From Heun Class Equations to Painlevé Equations 39
4.15 From Heun
(
; 7
2
)
to Painlevé I
Set
σ(z) = 1, τ(z) = 0, η(z) = −4z3 − 2tz − c.
Heun
(
; 7
2
)
equation:
∂2z − 4z3 − 2tz − c.
Deformed Heun
(
; 7
2
)
equation:
∂2z −
1
(z − λ)
∂z − 4z3 − 2tz + 4λ3 + 2tλ− µ2 +
µ
(z − λ)
.
Type Bq compatibility functions scaled with ε = −6:
a(z) :=
1
2(z − λ)
, b(z) = − µ
2(z − λ)
.
Painlevé
(
; 7
2
)
Hamiltonian:
H =
1
2
µ2 − 2λ3 − tλ. (4.43)
Painlevé
(
; 7
2
)
equation:
d2λ
dt2
= 6λ2 + t. (4.44)
In the standard terminology (4.44) is called the Painlevé I equation.
5 Five supertypes of Painlevé equation
5.1 Overview of five supertypes
Recall that the ten types of the Heun class equation can be grouped into five supertypes,
as described in Section 3.6. The ten types of Painlevé equation can be also grouped into five
supertypes. There is an exact correspondence between the supertypes of Heun class and Painlevé
equations:
� Painlevé VI or (1111).
� Painlevé V or (112).
– Painlevé ndeg-V or (112).
– Painlevé deg-V or
(
113
2
)
.
� Painlevé III′ or (22).
– Painlevé ndeg-III′ or (22).
– Painlevé deg-III′ or
(
3
22
)
.
– Painlevé ddeg-III′ or
(
3
2
3
2
)
.
� Painlevé IV-34 or (13).
– Painlevé IV or (13).
40 J. Dereziński, A. Ishkhanyan and A. Latosiński
– Painlevé 34 or
(
15
2
)
.
� Painlevé I–II or (4).
– Painlevé II or (4).
– Painlevé I or
(
7
2
)
.
In what follows we discuss this classification. We describe the minimal set of parameters that
can be used in a given type and various equivalences. In our discussion we try to include the
Hamiltonian aspect, whenever it is possible.
The above classification of Painlevé equation was pointed out by Ohyama–Okumura, see the
beginning of Section 2 of [19]. (In that reference the authors use the word “type” both for what
we call “supertype” and “type”.) The discussion in [19], however, concentrated on the second
order equations. Less space was devoted to the Hamiltonian form of the five supertypes.
The first supertype is Painlevé VI or (1111), which contains only one type. All the four
remaining supertypes contain at least two types. We discuss them in the following subsections.
In each of the following subsections we start with a general form of the given supertype of
the Painlevé equation. It will be indicated by �. It always has the form
d2λ
dt2
= F
(
t,
dλ
dt
, λ
)
.
The corresponding differential (nonlinear) operator will be denoted
P (t, λ) :=
d2λ
dt2
− F
(
t,
dλ
dt
, λ
)
.
We also introduce the corresponding Hamiltonian H(t, λ, µ). Both P (t, λ) and H(t, λ, µ) depend
on several parameters, put as subscripts. Next we give the scaling properties of the equation
and the Hamiltonian.
Then we list various nontrivial types that belong to a given supertype, marking them with ∗.
Finally, we discuss the relationship between various types. In particular, show how to reduce
the number of parameters using scaling.
Each supertype contains one generic type, which we call non-degenerate. Besides, it may
contain one or more degenerate types. The Hamiltonian that covers the non-degenerate type,
does not always allow us to describe all types that belong to a given supertype. This can be
viewed as a drawback of the Hamiltonian approach.
5.2 Painlevé V or (112)
As noted in [19], the usual form of the Painlevé V equation, depending on 4 parameters, should
be treated not as a single type, but as a supertype. It is invariant with respect to a scaling
transformation. It includes two nontrivial types: nondegenerate V depending on 3 parameters
and degenerate V depending on 2 parameters. There exists also a trivial type, solvable in
quadrature.
In this subsection we discuss the supertype Painlevé V in detail. Note that we prefer to
denote it by (112), since it corresponds to the supertype (112) of the Heun class.
� Painlevé V or (112) equation and Hamiltonian
d2λ
dt2
=
(
1
2λ
+
1
λ− 1
)(
dλ
dt
)2
− 1
t
dλ
dt
+
(λ− 1)2
t2
(
αλ+
β
λ
)
+ γ
λ
t
+ δ
λ(λ+ 1)
λ− 1
, (5.1)
tH = (λ− 1)2λµ2 −
(
κ0(λ− 1)2 + (χ1 − 1)λ(λ− 1)− ηtλ
)
µ
From Heun Class Equations to Painlevé Equations 41
+
(
(κ0 + χ1 − 1)2 − κ2∞
)
(λ− 1)
4
, (5.2)
where α = 1
2κ
2
∞, β = −1
2κ
2
0, γ = χ1η, δ = −1
2η
2.
Scaling properties
ε2Pα,β,γ,δ(εt, λ) = Pα,β,εγ,ε2δ(t, λ),
εHκ0,κ∞,χ1,η(εt, λ, µ) = Hκ0,κ∞,χ1,
η
ε
(t, λ, µ).
∗ Painlevé ndeg-V or (211) equation and Hamiltonian recalled from (4.26) and (4.25):
d2λ
dt2
=
(
1
2λ
+
1
λ− 1
)(
dλ
dt
)2
− 1
t
dλ
dt
+
(λ− 1)2
t2
(
αλ+
β
λ
)
+ γ
λ
t
− λ(λ+ 1)
2(λ− 1)
,
tH = (λ− 1)2λµ2 −
(
κ0(λ− 1)2 + (χ1 − 1)λ(λ− 1)− tλ
)
µ
+
(
(κ0 + χ1 − 1)2 − κ2∞
)
(λ− 1)
4
,
where α = 1
2κ
2
∞, β = −1
2κ
2
0, γ = χ1.
∗ Painlevé deg-V or
(
3
211
)
equation and Hamiltonian, recalled from (4.28) and (4.27):
d2λ
dt2
=
(
1
2λ
+
1
λ− 1
)(
dλ
dt
)2
− 1
t
dλ
dt
+
(λ− 1)2
t2
(
αλ+
β
λ
)
− 2
λ
t
,
tH = λ(λ− 1)2µ2 − κ0(λ− 1)2µ+
(κ20 − κ2∞)(λ− 1)
4
− tλ
(λ− 1)
, (5.3)
where α = 1
2κ
2
∞, β = −1
2κ
2
0.
Let us discuss special cases:
� Let δ 6= 0. In the Hamiltonian form it corresponds to η 6= 0. By scaling we can set
δ = −1
2 , and in the Hamiltonian form η = 1. We obtain the Painlevé (112) equation and
Hamiltonian.
� Let δ = 0, γ 6= 0. By scaling we can set γ = −2. We obtain the Painlevé
(
113
2
)
equation.
However, on the Hamiltonian level this reduction does not work: we cannot directly
use (5.2) to obtain the Painlevé
(
113
2
)
Hamiltonian.
� Let δ = 0, γ = 0. On the Hamiltonian level, η = 0. The Hamiltonian becomes
tH = (λ− 1)2λµ2 −
(
κ0(λ− 1)2 + (χ1 − 1)λ(λ− 1)
)
µ
+
(
(κ0 + χ1 − 1)2 − κ2∞
)
(λ− 1)
4
. (5.4)
This case is solvable in quadratures by the method of Section B.3.
Note that the corresponding 2nd order equation does not depend on the parameter χ1. On
the Hamiltonian level it can be seen by using the canonical transformation µ̃ = µ− χ1−1
2(λ−1) ,
which transforms (5.4) into
tH = (λ− 1)2λµ̃2 − κ0(λ− 1)2µ̃+
(κ20 − κ2∞)(λ− 1)
4
− (χ1 − 1)2
4
,
where the dependence on χ1 remains only in the free term.
42 J. Dereziński, A. Ishkhanyan and A. Latosiński
Remark 5.1. It is well known that the Painlevé deg-V or
(
113
2
)
and ndeg-III′ or (22) equations
are equivalent [19]. Below we will show this by describing a canonical transformation that
connects the corresponding Hamiltonians.
Let us insert the canonical transformation
λ̃ = 1− 1
µ
, µ =
1
1− λ̃
, µ̃ = µ2λ− (χ0 − 1)µ
2
, λ =
(
1− λ̃
)2
µ̃+
(χ0 − 1)
2
(
1− λ̃
)
,
into the (22) Hamiltonian (5.6). We obtain
tH = λ̃
(
λ̃− 1
)2
µ̃2+
−χ0+ χ∞+ 1
2
(
λ̃− 1
)2
µ̃− χ∞(χ0− 1)
2
(
λ̃− 1
)
− tλ̃(
λ̃− 1
)− (χ0− 1)2
4
+ t,
which after appropriate identification of parameters coincides with the
(
113
2
)
Hamiltonian (5.3)
up to a free term.
5.3 Painlevé III′ or (22)
As noted in [19], the usual Painlevé III′ equation, depending on 4 parameters, should be treated
as a supertype. It is invariant with respect to two distinct scaling transformations. It includes 3
nontrivial types: nondegenerate III′ depending on 2 parameters, degenerate III′ (in two forms)
depending on 1 parameter, doubly degenerate III′ with no parameters. There are also trivial
forms solvable in quadratures.
In this subsection we discuss Painlevé III′ in detail. We prefer to denote it by the symbol (22),
because it corresponds to the supertype (22) of the Heun class.
� Painlevé III′ or (22) equation and Hamiltonian:
d2λ
dt2
=
1
λ
(
dλ
dt
)2
− 1
t
dλ
dt
+
αλ2 + γλ3
4t2
+
β
4t
+
δ
4λ
,
tH = λ2µ2 −
(
η∞λ
2 + (χ0 − 1)λ− η0t
)
µ+
1
2
η∞(χ0 + χ∞ − 1)λ, (5.5)
where α = −4η∞χ∞, β = 4η0χ0, γ = 4η2∞, δ = −4η20. Scaling properties
ε2
ω
Pα,β,γ,δ(εt, ωλ) = P
ωα, ε
ω
β,ω2γ, ε
2
ω2
δ
(t, λ),
εHη0,η∞,χ0,χ∞
(
εt, ωλ,
µ
ω
)
= H ε
ω
η0,ωη∞,χ0,χ∞(t, λ, µ).
∗ Painlevé ndeg-III′ or (22) equation and Hamiltonian, recalled from (4.29) and (4.30):
d2λ
dt2
=
1
λ
(
dλ
dt
)2
− 1
t
dλ
dt
+
αλ2
4t2
+
λ3
t2
+
β
4t
− 1
λ
,
tH := λ2µ2 −
(
λ2 + (χ0 − 1)λ− t
)
µ+
1
2
(χ0 + χ∞ − 1)λ, (5.6)
where α = −4χ∞, β = 4χ0.
∗ Painlevé deg-III′-1 or
(
2; 3
2
)
equation and Hamiltonian, recalled from (4.32) and (4.31):
d2λ
dt2
=
1
λ
(
dλ
dt
)2
− 1
t
dλ
dt
− λ2
t2
+
β
4t
− 1
λ
,
tH = λ2µ2 +
(
(1− χ0)λ+ t
)
µ+
λ
2
,
where β = 4χ0.
From Heun Class Equations to Painlevé Equations 43
∗ Painlevé deg-III′-2 or
(
3
2 ; 2
)
equation and Hamiltonian recalled from (4.34) and (4.33):
d2λ
dt2
=
1
λ
(
dλ
dt
)2
− 1
t
dλ
dt
+
αλ2
4t2
+
λ3
t2
+
1
t
,
tH = λ2µ2 − λ2µ+
χ∞λ
2
+
t
2λ
,
where α = −4χ∞.
∗ Painlevé ddeg-III′ or
(
3
2
3
2
)
equation and Hamiltonian, recalled from (4.36) and (4.35):
d2λ
dt2
=
1
λ
(
dλ
dt
)2
− 1
t
dλ
dt
− λ2
t2
+
1
t
,
tH = λ2µ2 + λµ+
λ
2
+
t
2λ
.
Let us discuss special cases:
� Let δ 6= 0, γ 6= 0. On the level of the Hamiltonian it means η0 6= 0, η∞ 6= 0. By scaling
we can set on the Hamiltonian level η0 = 1, η∞ = 1, which corresponds to γ = 4, δ = −4.
We obtain the Painlevé ndeg-III′ or (22) equation (4.30) and Hamiltonian (4.29).
� Let δ, α 6= 0, γ = 0. By scaling we can make δ = −4, α = −4. We obtain the Painlevé(
2; 3
2
)
equation. This reduction does not work for the Painlevé
(
2; 3
2
)
Hamiltonian.
� Let δ = 0, γ, β 6= 0. By scaling we can make γ = 4, β = 4. We obtain the Painlevé
(
3
2 ; 2
)
equation. This reduction does not work for the Painlevé
(
3
2 ; 2
)
Hamiltonian.
The equations
(
3
2 ; 2
)
and
(
2; 3
2
)
are two equivalent forms of
(
23
2
)
see Proposition 5.2.
� Let δ = γ = 0, α, β 6= 0. By scaling we can make α = −4, β = 4. We obtain the
Painlevé
(
3
2
3
2
)
equation (4.36). This reduction does not work for the Painlevé
(
3
2
3
2
)
Hamil-
tonian (4.35).
� Let α = γ = 0. On the Hamiltonian level, η∞ = 0. The Hamiltonian is
tH = λ2µ2 −
(
(χ0 − 1)λ− η0t
)
µ. (5.7)
We can apply to (5.7) the time-dependent canonical transformation
H̃ = H − λ̃µ̃
t
, λ̃ =
λ
t
, µ̃ = tµ, (5.8)
obtaining
tH̃ = λ̃2µ̃2 −
(
χ0λ̃− η0
)
µ̃.
It is solvable by quadratures by Section B.3.
� Let β = δ = 0. On the Hamiltonian level it corresponds to η0 = 0. The Hamiltonian is
tH = λ2µ2 −
(
η∞λ
2 + (χ0 − 1)λ
)
µ+
1
2
η∞(χ0 + χ∞ − 1)λ.
It is solvable by quadratures by Section B.3.
Proposition 5.2. The Painlevé
(
2; 3
2
)
and
(
3
2 ; 2
)
equations are equivalent.
44 J. Dereziński, A. Ishkhanyan and A. Latosiński
Proof. First we apply to the Painlevé
(
2; 3
2
)
Hamiltonian the time-dependent canonical trans-
formation (5.8) obtaining
tH̃ = λ̃2µ̃2 +
(
−χ0λ̃+ 1
)
µ̃+
tλ̃
2
.
Next we apply the time independent canonical transformation
λ̃ =
1
λ
, µ̃ = −µλ2 +
χ0λ
2
,
obtaining
tH̃ = λ2µ2 − λ2µ+
χ0λ
2
+
t
2λ
− χ2
0
4
,
which is the Painlevé
(
3
2 ; 2
)
Hamiltonian for χ∞ = χ0 minus
χ2
0
4 . �
Remark 5.3. The standard Painlevé III equation is given by
d2λ̃
dt̃2
=
1
λ
(
dλ̃
dt̃
)2
− 1
t̃
dλ̃
dt̃
+
αλ̃2 + β
t̃
+ γλ̃3 +
δ
λ̃
. (5.9)
The Painlevé III′ equation is obtained from (5.9) by λ̃ = tλ, t̃ = t2.
5.4 Painlevé IV-34 or (13)
It has been noted in [19] that it is natural to consider Painlevé IV together with the so-called
Painlevé 34. The latter is equivalent to Painlevé II, and therefore is not so well known. Together
they can be treated as special cases of a supertype, which in [19] is denoted 4 34, and we denote
by IV-34, or preferably by (13), since it is related to the supertype (13) of the Heun class. In this
subsection we discuss this supertype of Painlevé in detail.
Painlevé IV-34 depends on 3 parameters. It is invariant with respect to a scaling transfor-
mation. It contains Painlevé IV depending on 2 parameters and Painlevé 34 depending on 1
parameter, as well as a trivial type solvable in quadratures.
� Painlevé IV-34 or (13) equation and Hamiltonian:
d2λ
dt2
=
1
2λ
(
dλ
dt
)2
+ ρλ(2λ+ t) + γλ(λ+ t)(3λ+ t) +
β
4λ
,
H = λµ2 −
(
ηλ2 + ηtλ− θλ+ κ0
)
µ+
(
θ2
4
+
(κ0 − 1)η
2
)
λ, (5.10)
where β = −2κ20, ρ = −ηθ, γ = 1
2η
2. Scaling properties
εPβ,ρ,γ(εt, ελ) = Pβ,ε3ρ,ε4γ(t, λ),
εHη,θ,κ0
(
εt, ελ, ε−1µ
)
= Hε2η,εθ,κ0(t, λ, µ).
∗ Painlevé IV or (13) equation and Hamiltonian, recalled from (4.38) and (4.37):
d2λ
dt2
=
1
2λ
(
dλ
dt
)2
+
3
2
λ3 + 4tλ2 + 2
(
t2 − α
)
λ+
β
λ
,
H = 2λµ2 −
(
λ2 + 2tλ+ 2κ0
)
µ+ θ∞λ,
where α = −κ0 + 2θ∞ − 1, β = −2κ20.
From Heun Class Equations to Painlevé Equations 45
∗ Painlevé 34 or
(
15
2
)
equation and Hamiltonian, recalled from (4.40) and (4.39):
d2λ
dt2
=
1
2λ
(dλ
dt
)2
+ 2λ2 + tλ+
β
4λ
,
H = λµ2 − κ0µ−
λ2
2
− tλ
2
, (5.11)
where β = −2κ20.
Let us discuss special cases:
� Let γ 6= 0. By scaling we can set γ = 1
8 . We change the time t̃ = 1
2 t + ρ. We obtain the
Painlevé (13) equation with α = 4ρ2.
The equivalent reduction on the Hamiltonian level: For η 6= 0, by scaling, we can make
η = 1
2 . We multiply the Hamiltonian by 2 and change t to 2t:
2H(2t, λ, µ) = 2λµ2 −
(
λ2 + 2tλ− 2θλ+ κ0
)
µ+
θ2 + κ0 − 1
2
λ,
which is the Painlevé (13) Hamiltonian with θ∞ = θ2+κ0−1
2 and t̃ = t− θ.
� Let γ = 0, ρ 6= 0. By scaling we can make ρ = 1. We obtain the Painlevé 34 or
(
15
2
)
equation. This reduction does not work for the Painlevé 34 or
(
15
2
)
Hamiltonian.
� Let γ = ρ = 0, which on the Hamiltonian level corresponds to η = 0. We have the
Hamiltonian
H = λµ2 − (−θλ+ κ0)µ+
θ2
4
λ,
which is solvable by quadratures (see Section B.3).
Remark 5.4. Note that Painlevé 34 or
(
15
2
)
is equivalent to Painlevé II or (4).
Let us show this on the Hamiltonian level. After an application of the canonical transfor-
mation
µ̃ = λ, λ̃ = −µ,
the Painlevé
(
15
2
)
Hamiltonian (5.11) becomes
H = − µ̃
2
2
− λ̃2µ̃− tµ̃
2
+ κ0λ̃.
Then we change t into −t and multiply the Hamiltonian by −1, obtaining the Painlevé (4)
Hamiltonian (5.12) with κ0 = α+ 1
2 .
5.5 Painlevé I–II or (4)
Usually Painlevé I and II equations are treated separately. However, it has been noted already by
Painlevé and elaborated in [19] that it is natural to join them in a single supertype. [19] denotes
it 1 2, we denote it I–II, or preferably (4), since it corresponds to the supertype (4) of the Heun
class. In this subsection we discuss this supertype of Painlevé in detail.
Painlevé I–II depends on 2 parameters. It is invariant with respect to a scaling transformation.
It contains Painlevé II, depending on 1 parameter, Painlevé I with no parameters and a trivial
type solvable in quadratures.
46 J. Dereziński, A. Ishkhanyan and A. Latosiński
� Painlevé I–II or (4) equation and Hamiltonian:
d2λ
dt2
= γ
(
2λ3 + tλ
)
+ β
(
6λ2 + t
)
,
H =
1
2
µ2 −
(
ηλ2 +
1
2
ηt
)
µ− 2βλ3 − tβλ− 1
2
ηλ,
where γ = η2.
Scaling properties:
ε3Pγ,β
(
ε2t, ελ
)
= Pε6γ,ε5β(t, λ),
ε2Hη,β
(
ε2t, ελ, ε−1µ
)
= Hε3η,ε5β(t, λ, µ).
∗ Painlevé II or (4) equation and Hamiltonian, recalled from (4.42) and (4.41):
d2λ
dt2
= 2λ3 + tλ+ α,
H =
1
2
µ2 −
(
λ2 +
t
2
)
µ−
(
α+
1
2
)
λ. (5.12)
∗ Painlevé I or
(
7
2
)
equation and Hamiltonian, recalled from (4.44) and (4.43):
d2λ
dt2
= 6λ2 + t,
H =
1
2
µ2 − 2λ3 − tλ.
Let us discuss special cases:
� Let η 6= 0, in both the equation and the Hamiltonian. By scaling we can set η = 1.
We apply the canonical transformation
t = t̃+ 6β2, λ = λ̃− β, µ = µ̃− 2βλ̃+ 4β2.
We obtain
H =
µ̃2
2
−
(
λ̃2 +
t̃
2
)
µ−
(
2β3 +
1
2
)
λ̃+
1
2
β − β2t̃.
Thus up to free terms we obtain the Painlevé (4) Hamiltonian with α = 2β3, and hence
also the Painlevé (4) equation.
� Let η = 0, β 6= 0. By scaling we can set β = 1. The Hamiltonian becomes
H =
µ2
2
− 2λ3 − tλ.
Thus we obtain the Painlevé
(
7
2
)
Hamiltonian, and hence also the Painlevé
(
7
2
)
equation.
� Let η = 0, β = 0. The Hamiltonian becomes H = 1
2µ
2, which is trivial.
From Heun Class Equations to Painlevé Equations 47
A Proof of Theorems 4.1, 4.2 and 4.3
A.1 Preparation for the proof of Theorem 4.1
Recall from (4.7), (4.8) and (4.12) that we set
p(z) := p0(z)−
1
z − λ
, p0(z) :=
τ(z)
σ(z)
,
q(z) := q0(z) +
1
σ(z)
(
−η(λ)− µ
(
τ(λ)− σ′(λ)
)
− µ2σ(λ) +
µσ(λ)
(z − λ)
)
, q0(z) :=
η(z)
σ(z)
,
a(z) :=
c(z)
z − λ
, b(z) := −c(λ)µ
z − λ
,
where c(z) is a t, λ-dependent polynomial of degree ≤ 2. Recall that the prime is synonymous
with ∂z and the dot with ∂t.
We will find conditions on c and the time variable t so that the compatibility conditions (4.5)
and (4.6), that is,
ṗ− ap′ + 2b′ − pa′ + a′′ = 0, (A.1)
q̇ + pb′ − 2qa′ − aq′ + b′′ = 0, (A.2)
are satisfied.
The following simple identities will be useful in our calculations:
Lemma A.1.
(λ− s)
(z − λ)
(
1
z − s
− 1
λ− s
+
z − λ
(λ− s)2
)
=
1
λ− s
− 1
z − s
,
2
z − s
− 2
λ− s
+ (z − λ)
(
1
(z − s)2
+
1
(λ− s)2
)
=
(z − λ)3
(z − s)2(λ− s)2
. (A.3)
Let ψ := ξ(z)
z−s . If ξ be a polynomial with deg ξ ≤ 2, then
ξ(z)− ξ(λ)− (z − λ)ξ′(z) = −ξ
′′
2
(z − λ)2,
ψ(z)− ψ(λ)− (z − λ)ψ′(z) = − ξ(s)(z − λ)2
(z − s)2(λ− s)
. (A.4)
If ξ is a polynomial with deg ξ ≤ 3, then
2ξ(z)− 2ξ(λ)− (z − λ)
(
ξ′(z) + ξ′(λ)
)
= −ξ
′′′
6
(z − λ)3, (A.5)
2ψ(z)− 2ψ(λ)− (z − λ)
(
ψ′(z) + ψ′(λ)
)
= ξ(s)
(z − λ)3
(z − s)2(λ− s)2
. (A.6)
A.2 First compatibility condition
Proposition A.2. Suppose that the following equation of motion for λ holds:
λ̇ = 2c(λ)µ− c′(λ) +
cτ
σ
(λ), (A.7)
and we have the condition
0 = I :=
τ̇
σ
(z)− τ σ̇
σ2
(z) +
cτ
σ (z)− cτ
σ (λ)− (z − λ)
(
cτ
σ
)′
(z)
(z − λ)2
. (A.8)
Then (A.1) is satisfied.
48 J. Dereziński, A. Ishkhanyan and A. Latosiński
Proof.
0 = ṗ(z)− a(z)p′(z) + 2b′(z)− p(z)a′(z) + a′′(z)
= ṗ0(z)−
λ̇
(z − λ)2
− c(z)
z − λ
(
p′0(z) +
1
(z − λ)2
)
+
2c(λ)µ
(z − λ)2
−
(
p0(z)−
1
(z − λ)
)(
c′(z)
z − λ
− c(z)
(z − λ)2
)
+
c′′(z)
z − λ
− 2c′(z)
(z − λ)2
+
2c(z)
(z − λ)3
=
1
(z − λ)2
(
−λ̇+ 2c(λ)µ− c′(z) + (cp0)(z)
)
+
1
(z − λ)
(
−c(z)p′0(z)− p0(z)c′(z) + c′′(z)
)
+ ṗ0(z).
We rearrange this as
=
1
(z − λ)2
(
−λ̇+ 2c(λ)µ− c′(λ) + (cp0)(λ)
)
(A.9)
+
−c′(z) + c′(λ) + c′′(z)(z − λ)
(z − λ)2
(A.10)
+
(cp0)(z)− (cp0)(λ)− (z − λ)
(
cp0)
′(z)
(z − λ)2
+ ṗ0(z). (A.11)
(A.9) is proportional to 1
(z−λ)2 and the last two lines are regular at z = λ. Therefore, (A.9) has
to vanish separately, yielding the condition (A.7). (A.10) vanishes automatically, because c is
a polynomial of degree ≤ 2 in z. (A.11) yields the condition (A.8). �
A.3 Second compatibility condition
It is much more difficult to analyze the second compatibility condition.
Proposition A.3. Suppose that the equation for λ (A.7) holds together with the equation for µ
µ̇ = −cη
′
σ
(λ)− µ
(
cτ ′
σ
(λ)− cσ′′
2σ
(λ)− c′′
2
)
− µ2σ
′c
σ
(λ). (A.12)
Assume also that I = 0 (see (A.8)) and the following conditions are satisfied:
0 = II := − σ̇
σ
(z)
(
η(z)− η(λ)
)
+ η̇(z)− η̇(λ)
+
η(z)− η(λ)
(z − λ)
(
cσ′
σ
(z)− c′(z)
)
− η′(λ)
(
σ′c
σ
(λ)− c′(λ)
)
+
2cη(z)− 2cη(λ)−
(
(cη)′(z) + (cη)′(λ)
)
(z − λ)
(z − λ)2
;
0 = III :=
σ̇
σ
(z)− σ̇
σ
(λ)−
cσ′
σ (z)− cσ′
σ (λ)− ( cσ
′
σ )′(λ)(z − λ)
(z − λ)
.
Then (A.2) is true.
Proof.
0 = q̇(z) + p(z)b′(z)− 2q(z)a′(z)− a(z)q′(z) + b′′(z)
=
1
σ(z)
(
−τ(λ) + σ′(λ)− 2µσ(λ) +
σ(λ)
(z − λ)
)
µ̇
From Heun Class Equations to Painlevé Equations 49
+
1
σ(z)
(
−η′(λ)− µ(τ ′(λ)− σ′′(λ))− µ2σ′(λ) +
µσ′(λ)
(z − λ)
+
µσ(λ)
(z − λ)2
)
λ̇
− σ̇(z)
σ2(z)
(
η(z)− η(λ)− µ(τ(λ)− σ′(λ))− µ2σ(λ) +
µσ(λ)
(z − λ)
)
+
1
σ(z)
(
η̇(z)−η̇(λ)− µ(τ̇(λ)− σ̇′(λ))− µ2σ̇(λ)+
µσ̇(λ)
(z−λ)
)
+
(
τ(z)
σ(z)
− 1
(z−λ)
)
c(λ)µ
(z−λ)2
− 2
σ(z)
(
η(z)− η(λ)− µ
(
τ(λ)− σ′(λ)
)
− µ2σ(λ) +
µσ(λ)
(z − λ)
)(
c′(z)
(z − λ)
− c(z)
(z − λ)2
)
+
c(z)σ′(z)
(z − λ)σ2(z)
(
η(z)− η(λ)− µ
(
τ(λ)− σ′(λ)
)
− µ2σ(λ) +
µσ(λ)
(z − λ)
)
− c(z)
(z − λ)σ(z)
(
η′(z)− µσ(λ)
(z − λ)2
)
− 2
c(λ)µ
(z − λ)3
.
Next we collect the terms that contain an inverse power of z − λ. These terms are grouped
in several categories. In these terms we also insert (A.7). We obtain
=
1
σ(z)
(
−τ(λ)+ σ′(λ)− 2µσ(λ)
)
µ̇+
1
σ(z)
(
−η′(λ)− µ(τ ′(λ)− σ′′(λ))− µ2σ′(λ)
)
λ̇
− σ̇(z)
σ2(z)
(
η(z)− η(λ)− µ(τ(λ)− σ′(λ))− µ2σ(λ)
)
+
1
σ(z)
(
η̇(z)− η̇(λ)− µ(τ̇(λ)− σ̇′(λ))− µ2σ̇(λ)
)
+
1
(z − λ)σ(z)
(
−µσ̇(z)σ(λ)
σ(z)
+ µσ̇(λ) + µ̇σ(λ)
)
+
1
(z − λ)2σ(z)
2c(z)
(
η(z)− η(λ)
)
+
1
(z − λ)σ(z)
(
−2c′(z)
(
η(z)− η(λ)
)
+ c(z)
σ′(z)
σ(z)
(
η(z)− η(λ)
)
− c(z)η′(z)
)
+
µ
(z − λ)2σ(z)
(
c(λ)τ(λ) + c(λ)τ(z)− 2c(z)τ(λ)
)
+
µ
(z − λ)σ(z)
(
c(λ)
σ′(λ)
σ(λ)
τ(λ)− c(z)σ
′(z)
σ(z)
τ(λ) + 2c′(z)τ(λ)
)
+
µ
(z − λ)3σ(z)
(
3c(z)σ(λ)− 3c(λ)σ(z)
)
+
µ
(z − λ)2σ(z)
(
−c′(λ)σ(λ)− 2c′(z)σ(λ) + 2c(z)σ′(λ) + c(z)
σ′(z)
σ(z)
σ(λ)
)
+
µ
(z − λ)σ(z)
(
−c′(λ)σ′(λ)− 2c′(z)σ′(λ) + c(z)
σ′(z)
σ(z)
σ′(λ)
)
+
µ2
(z − λ)2σ(z)
(
2c(λ)σ(λ)− 2c(z)σ(λ)
)
+
µ2
(z − λ)σ(z)
(
2c(λ)σ′(λ) + 2c′(z)σ(λ)− c(z)σ
′(z)
σ(z)
σ(λ)
)
= − 1
σ(z)
(
η′(λ) + µ
(
τ ′(λ)− σ′′(λ)
)
+ µ2σ′(λ)
)
λ̇− 1
σ(z)
(
τ(λ)− σ′(λ) + 2µσ(λ)
)
µ̇
+
σ(λ)
σ(z)(z − λ)
µ̇+
1
σ(z)
(
− σ̇
σ
(z)
(
η(z)− η(λ)
)
+ η̇(z)− η̇(λ)
)
+
µ
σ(z)
(
(λ)
σ̇
σ
(z)− τ̇(λ)
)
+
µ
σ(z)
(
−σ′(λ)
σ̇
σ
(z) + σ̇′(λ)− σ(λ)
(
σ̇
σ (z)− σ̇
σ (λ)
)
(z − λ)
)
+
µ2
σ(z)
σ(λ)
(
σ̇
σ
(z)− σ̇
σ
(λ)
)
+
1
(z − λ)σ(z)
c(λ)η′(λ)
50 J. Dereziński, A. Ishkhanyan and A. Latosiński
+
1
σ(z)
(
2cη(z)−2cη(λ)−
(
(cη)′(z)+(cη)′(λ)
)
(z−λ)
(z−λ)2
+
η(z)−η(λ)
(z−λ)
(
c
σ′
σ
(z)−c′(z)
))
+
µ
σ(z)(z − λ)
c(λ)τ ′(λ) +
µ
σ(z)
(
(σ
′c
σ (λ)− σ′c
σ (z))
(z − λ)
τ(λ) + c(λ)
τ ′′
2
+ c′′τ(λ)
)
+
µ
(z − λ)σ(z)
(
−c
′′σ(λ)
2
− c(λ)σ′′(λ)
2
)
+
µ
σ(z)
(
−c′′σ′(λ) +
(
cσ′
σ (z)− cσ′
σ (λ)− (z − λ)
(
cσ′
σ
)′
(λ)
)
(z − λ)2
σ(λ)
+
(
cσ′
σ (z)− cσ′
σ (λ)
)
z − λ
σ′(λ)− σ′′′c(λ)
2
)
+
µ2
σ(z)(z − λ)
c(λ)σ′(λ) +
µ2
σ(z)
(
c′′σ(λ)− σ(λ)
(
σ′c
σ (z)− σ′c
σ (λ)
)
(z − λ)
)
.
The singular term equals
1
σ(z)(z − λ)
(
σ(λ)µ̇+ c(λ)η′(λ) + µ
(
c(λ)τ ′(λ)− 1
2
c(λ)σ′′(λ)− 1
2
c′′σ(λ)
)
+ µ2c(λ)σ′(λ)
)
.
It yields the equation for µ̇, that is (A.12). After inserting (A.7) and (A.12) the first two lines
become
1
σ(z)σ(λ)
η′(λ)
(
−c(λ)σ′(λ) + c′(λ)σ(λ)
)
+
µ
σ(z)σ(λ)
(
τ ′(λ)
(
−c(λ)σ′(λ) + c′(λ)σ(λ)
)
+ τ(λ)
(
−c
′′σ(λ)
2
+
c(λ)σ′′(λ)
2
))
+
µ
σ(z)σ(λ)
(
c′′σ′(λ)σ(λ)
2
+
c(λ)σ′(λ)σ′′(λ)
2
− c′(λ)σ(λ)σ′′(λ)
)
+
µ2
σ(z)σ(λ)
(
−c(λ)σ′(λ)2 + c′(λ)σ(λ)σ′(λ)− c′′σ(λ)2 + c(λ)σ′′(λ)σ(λ)
)
.
Finally, we obtain
=
1
σ(z)
(
− σ̇
σ
(z)
(
η(z)− η(λ)
)
+ η̇(z)− η̇(λ) +
η(z)− η(λ)
(z − λ)
(
cσ′
σ
(z)− c′(z)
)
− η′(λ)
(
σ′c
σ
(λ)− c′(λ)
)
+
2cη(z)− 2cη(λ)−
(
(cη)′(z) + (cη)′(λ)
)
(z − λ)
(z − λ)2
)
+
µ
σ(z)
(
−τ̇(λ) + c(λ)
τ ′′
2
− τ ′(λ)
(
σ′c
σ
(λ)− c′(λ)
)
+ τ(λ)
(
σ̇
σ
(z) +
c′′
2
+
c(λ)σ′′(λ)
2σ(λ)
−
(
σ′c
σ (z)− σ′c
σ (λ)
)
(z − λ)
))
+
µ
σ(z)
(
−σ′(λ)
σ̇
σ
(z) + σ̇′(λ)− σ(λ)
(
σ̇
σ (z)−
(
σ̇
σ (λ)
)
(z − λ)
− c′′
2
σ′(λ) +
c(λ)
2
σ′(λ)
σ(λ)
σ′′(λ)− c′(λ)σ′′(λ)− σ′′′c(λ)
2
+
(
cσ′
σ (z)− cσ′
σ (λ)− (z − λ)
(
cσ′
σ
)′
(λ)
)
(z − λ)2
σ(λ) +
(
cσ′
σ (z)− cσ′
σ (λ)
)
z − λ
σ′(λ)
)
From Heun Class Equations to Painlevé Equations 51
+
µ2σ(λ)
σ(z)
(
σ̇
σ
(z)− σ̇
σ
(λ)−
cσ′
σ (z)− cσ′
σ (λ)− ( cσ
′
σ )′(λ)(z − λ)
(z − λ)
)
=
1
σ(z)
II +
(
− µσ(λ)
σ(z)(z − λ)
+
µ
σ(z)
(
τ(λ)− σ′(λ)
)
+
µ2σ(λ)
σ(z)
)
III
+
µ
σ(z)
(
(τ(λ)− σ′(λ))
σ̇
σ
(λ)− τ̇(λ) + σ̇′(λ)
)
+
µ
σ(z)
(
1
2
(
(τ − σ′)c
)′′
(λ)− (τ − σ′)′(λ)
cσ′
σ
(λ)
+ (τ − σ′)(λ)
(
−c
′σ′
σ
(λ) +
cσ′σ′
σ2
(λ)− cσ′′
2σ
(λ)
))
=
1
σ(z)
II +
(
− µσ(λ)
σ(z)(z − λ)
+
µ
σ(z)
(
τ(λ)− σ′(λ)
)
+
µ2σ(λ)
σ(z)
)
III
+
µσ(λ)
σ(z)
(
−∂t
τ
σ
(λ) + ∂t
σ′
σ
(λ) +
1
2
(τc
σ
)′′
(λ)− 1
2
(σ′c
σ
)′′
(λ)
)
. (A.13)
We have
lim
z→λ
I = ∂t
τ
σ
(λ)− 1
2
(τc
σ
)′′
(λ), (A.14)
lim
z→λ
III
(z − λ)
= ∂t
σ′
σ
(λ)− 1
2
(σ′c
σ
)′′
(λ). (A.15)
Therefore, if I and III vanish, then so do (A.14) and (A.15), and hence also (A.13) �
Propositions A.2 and A.3 prove Theorem 4.1.
Next we would like to prove Theorems 4.2 and 4.3. The proof will be divided into three
subsections. In the first two we consider Cases A and in the third Case B.
From now on we assume that σ, τ , η correspond to a Heun class equation, that is deg σ ≤ 3,
deg τ ≤ 2 and deg ση ≤ 4.
A.4 Case A, Part I
Assume that σ has a zero at z = s so that σ(z) = (z − s)ρ(z). Clearly, deg ρ ≤ 2.
We make the ansatz
c(z) := m(λ− s)ρ(z), (A.16)
where m is a function only of t.
The equations (A.7) and (A.12) can be rewritten as
λ̇ = m
(
2σ(λ)µ− (λ− s)ρ′(λ) + τ(λ)
)
, (A.17)
µ̇ = −m
(
η′(λ) + µ(τ ′(λ)− ρ′(λ)− (λ− s)ρ′′(λ)) + µ2σ′(λ)
)
. (A.18)
It is easy to check that (A.17) and (A.18) are the Hamilton equations for the Hamiltonian
H = m
(
η(λ) +
(
τ(λ)− (λ− s)ρ′(λ)
)
µ+ σ(λ)µ2
)
.
Another consequence of (A.16) is
cτ
σ
(z) =
m(λ− s)τ(z)
(z − s)
.
52 J. Dereziński, A. Ishkhanyan and A. Latosiński
Therefore, using (A.4) we obtain
cτ
σ (z)− cτ
σ (λ)− (z − λ)
(
cτ
σ
)′
(z)
(z − λ)2
= − mτ(s)
(z − s)2
.
Hence,
I = ∂t
τ
σ
(z)− mτ(s)
(z − s)2
.
Using
cσ′
σ
(z)− c′(z) =
m(λ− s)ρ(z)
z − s
,
we obtain
η(z)− η(λ)
(z − λ)
(
cσ′
σ
(z)− c′(z)
)
− η′(λ)
(
σ′c
σ
(λ)− c′(λ)
)
=
m
(
η(z)− η(λ)
)
(λ− s)ρ(z)
(z − λ)(z − s)
−mη′(λ)ρ(λ).
Therefore,
II = − σ̇
σ
(z)
(
η(z)− η(λ)
)
+ η̇(z)− η̇(λ) +
m
(
η(z)− η(λ)
)
(λ− s)ρ(z)
(z − λ)(z − s)
−mη′(λ)ρ(λ)
+
m(λ− s)
(z − λ)2
(
2ρη(z)− 2ρη(λ)−
(
(ρη)′(z) + (ρη)′(λ)
)
(z − λ)
)
.
Finally, we have
cσ′(z)
σ(z)
=
m(λ− s)(ρ(z) + (z − s)ρ′(z))
(z − s)
.
Using (A.4) with ξ(z) = −ρ(z)− (z − s)ρ′(z) and the interchanged role of λ and z we obtain
cσ′
σ (z)− cσ′
σ (λ)− ( cσ
′
σ )′(λ)(z − λ)
(z − λ)
= mρ(s)
(z − λ)
(z − s)(λ− s)
.
Therefore,
III =
σ̇
σ
(z)− σ̇
σ
(λ)−mρ(s)
(z − λ)
(z − s)(λ− s)
.
Subcase A1. We assume that the root s is single and the time variable is chosen as t = s.
We can write σ(z) = (z − t)ρ(z), ρ(t) 6= 0, deg ρ ≤ 2.
We assume deg η0 ≤ 1, deg φ ≤ 1, κ, α ∈ C,
τ(z) = (1− κ)ρ(z) + φ(z)(z − t), η(z) =
αρ(t)
z − t
+ η0(z), κ̇ = ρ̇ = φ̇ = η̇0 = α̇ = 0.
We have
∂t
τ
σ
(z) =
1− κ
(z − t)2
, 1− κ =
τ(t)
ρ(t)
.
From Heun Class Equations to Painlevé Equations 53
Therefore,
I =
1− κ
(z − t)2
(
1−mρ(t)
)
.
Using deg η0 ≤ 1 and η̇0 = 0 we obtain
− σ̇
σ
(z)
(
η0(z)− η0(λ)
)
+ η̇0(z)− η̇0(λ) =
1
(z − t)
η′0(z − λ),
m
(
η0(z)− η0(λ)
)
(λ− t)ρ(z)
(z − λ)(z − t)
−mη′0(λ)ρ(λ) = −mρ(t)η′0(z − λ)
(z − t)
+mη′0(λ− t)
ρ′′
2
(z − λ).
Using deg η0ρ ≤ 3 and (A.5) we get
m(λ− t)
(z − λ)2
(
2ρη0(z)− 2ρη0(λ)−
(
(ρη0)
′(z) + (ρη0)
′(λ)
)
(z − λ)
)
= −m(λ− t)(ρη0)′′′(z − λ)
6
= −m(λ− t)ρ′′η′0(z − λ)
2
.
Set ψ(z) := ρ(t)
z−t . We have
− σ̇
σ
(z)
(
ψ(z)− ψ(λ)
)
+ ψ̇(z)− ψ̇(λ) =
(
ρ(t)
z − t
+ ρ′(t)
)(
1
z − t
− 1
λ− t
)
+ ρ(t)
(
1
(z − t)2
− 1
(λ− t)2
)
,
m
(
ψ(z)− ψ(λ)
)
(λ− t)ρ(z)
(z − λ)(z − t)
−mψ′(λ)ρ(λ) = − mρ(t)
(z − t)2
(
ρ(t) + ρ′(t)(z − t) +
ρ′′
2
(z − t)2
)
+
mρ(t)
(λ− t)2
(
ρ(t)+ ρ′(t)(λ− t)+
ρ′′
2
(λ− t)2
)
.
Using (A.6) we get
m(λ− t)
(z − λ)2
(
2ρψ(z)− 2ρψ(λ)−
(
(ρψ)′(z) + (ρψ)′(λ)
)
(z − λ)
)
=
mρ(t)2(z − λ)
(z − t)2(λ− t)
= −mρ(t)2
(z − t)
(
1
z − t
− 1
λ− t
)
.
Combining the above identities we obtain
II =
((
−(λ− t)η′0 +
ρ(t)
z − t
+ ρ′(t)
)(
1
z − t
− 1
λ− t
)
+ ρ(t)
(
1
(z − t)2
− 1
λ− t)2
))
× (1−mρ(t)
)
.
We have
σ̇
σ
(z)− σ̇
σ
(λ) = − 1
z − t
+
1
λ− t
=
z − λ
(λ− t)(z − t)
.
Therefore,
III =
z − λ
(λ− t)(z − t)
(
1−mρ(t)
)
.
Thus m = 1
ρ(t) implies I = II = III = 0.
54 J. Dereziński, A. Ishkhanyan and A. Latosiński
A.5 Case A, Part II
Assume that the root s is at least double. Then ρ(s) = 0 so that the normalization m = 1
ρ(s)
does not work. We have to change the time variable.
Thus we assume σ(z) = (z − s)2ρ1(z), where deg ρ1 ≤ 1. We also assume ṡ = ρ̇1 = 0. Then
we have
I =
τ̇
σ
(z)− mτ(s)
(z − s)2
,
II = η̇(z)− η̇(λ) +
m(λ− s)
(z − λ)
(
ρ1η(z)− ρ1η(λ)− (ρ1η)′(λ)(z − λ)− (ρη)′′′
6
(z − λ)2
)
.
We consider separately two subcases: in the first the time variable is contained in p0 and in the
second in q0.
Subcase Ap. We assume that τ̇0 = η̇ = 0, deg τ0 ≤ 2, ση(s) = (ση)′(s) = 0,
τ(z) = τ0(z) + tρ1(z), τ0(s) 6= 0 or ρ1(s) 6= 0.
We have
τ̇
σ
(z) =
1
(z − s)2
, τ(s) = τ0(s) + tρ1(s) 6≡ 0.
Hence,
I =
1
(z − s)2
(
1−m(τ0(s) + tρ1(s))
)
.
Thus m =
(
τ0(s) + tρ1(s)
)−1
implies I = 0.
ση(s) = (ση)′(s) = 0 implies that ρ1η is a polynomial of degree ≤ 2. Therefore, (ρη)′′′ =(
ρ1η)(z − s)
)′′′
= 3(ρ1η)′′ and
II =
m(λ− s)
(z − λ)
(
ρ1η(z)− ρ1η(λ)− (ρ1η)′(λ)(z − λ)− (ρ1η)′′
2
(z − λ)2
)
= 0.
Subcase Aq. We assume that τ(s) = 0, ση0(s) = 0, deg ση0 ≤ 4,
η(z) =
t
z − s
+ η0(z), σ̇ = τ̇ = η̇0 = 0,
ρ1(s) 6= 0 or (ση0)
′ 6= 0.
Clearly, I = 0.
Now
ση0(z) = (z − s)(ση0)′(s) + (z − s)2ψ(z),
where degψ ≤ 2. Therefore,
η0ρ1(z) =
(ση0)
′(s)
z − s
+ ψ(z),
ρ1η(z) =
(ση0)
′(s) + ρ1(s)t
z − s
+ ψ(z) + ρ′1t.
Now (ηρ)′′′ =
(
(z − s)ρ1η
)′′′
=
(
(z − s)ψ
)′′′
= 3ψ′′. Therefore,
ρ1η(z)− ρ1η(λ)− (ρ1η)′(λ)(z − λ)− (ρη)′′′
6
(z − λ)2
From Heun Class Equations to Painlevé Equations 55
= ψ(z)− ψ(λ)− (z − λ)ψ′(λ)− (z − λ)2
ψ′′
2
(A.19)
+
(
(ση0)
′(s) + ρ1(s)t
)( 1
z − s
− 1
λ− s
+
(z − λ)
(λ− s)2
)
. (A.20)
(A.19) vanishes, because degψ ≤ 2. Using (A.20), (A.3) and
η̇(z)− η̇(λ) =
1
z − s
− 1
λ− s
we obtain
II =
(
1
z − s
− 1
λ− s
)(
1−m
(
(ση0)
′(s) + ρ1(s)t
))
,
where (ση0)
′(s) + ρ1(s)t 6≡ 0. Hence if m =
(
(ση0)
′(s) + ρ1(s)t
)−1
, then II = 0.
A.6 Case B
We assume that deg σ ≤ 2, deg ση ≤ 3 and σ̇ = 0. We set
c(z) = mσ(z),
where m is a function just of t. The equations (A.7) and (A.12) can be rewritten as
λ̇ = m
(
2σ(λ)µ− σ′(λ) + τ(λ)
)
, (A.21)
µ̇ = −m
(
η′(λ) + µ(τ ′(λ)− σ′′(λ)) + µ2σ′(λ)
)
. (A.22)
We easily check that (A.21) and (A.22) are the Hamilton equations for the Hamiltonian
H(t, λ, µ) := m
(
η(λ) +
(
τ(λ)− σ′(λ)
)
µ+ σ(λ)µ2
)
.
Using deg τ ≤ 2, we get
τ(z)− τ(λ)− (z − λ)τ ′(z)
(z − λ)2
= −τ
′′
2
.
Therefore,
I =
τ̇(z)
σ(z)
−mτ ′′
2
.
Using deg ση ≤ 3 we obtain
2ση(z)− 2ση(λ)−
(
(ση)′(z) + (ση)′(λ)
)
(z − λ)
(z − λ)2
= −(ση)′′′
6
(z − λ).
Therefore,
II = η̇(z)− η̇(λ)−m(ση)′′′
6
(z − λ).
III is clearly 0, because σ̇ = 0 and cσ′
σ is a polynomial of degree ≤ 1.
This ends the proof of Theorem 4.3B.
Again, we consider two subcases, with t contained in p0 and in q0.
56 J. Dereziński, A. Ishkhanyan and A. Latosiński
Subcase Bp. deg σ ≤ 2, deg τ0 ≤ 2, deg ση ≤ 2,
τ(z) = tσ(z) + τ0(z), σ̇ = τ̇0 = η̇ = 0,
deg σ = 2 or deg τ0 = 2.
II = 0 is automatic. Moreover, τ ′′ = tσ′′ + τ ′′0 6≡ 0 and
I = 1−m
(
t
σ′′
2
+
τ ′′
2
)
.
Hence m =
(
tσ
′′
2 +
τ ′′0
2
)−1
implies I = 0.
Subcase Bq. deg σ ≤ 2, deg τ ≤ 1, deg ση0 ≤ 3,
η(z) = tz + η0(z), σ̇ = τ̇ = η̇0 = 0,
deg σ = 2 or deg ση0 = 3.
I = 0 is automatic. Moreover, (ση)′′′ = t3σ′′ + (ση)′′′ 6≡ 0 and
II = (z − λ)
(
1−m
(
t
σ′′
2
+
(ση0)
′′′
6
))
.
Therefore m =
(
tσ
′′
2 + (ση0)′′′
6
)−1
implies II = 0.
B Hamilton equations
B.1 From Hamilton equations to second order equations
We devote this appendix to a few remarks about Hamilton equations.
Suppose that H(t, λ, µ) is a time-dependent Hamiltonian. The equations
dλ
dt
=
∂H
∂µ
(t, λ, µ), (B.1)
dµ
dt
= −∂H
∂λ
(t, λ, µ), (B.2)
are called the Hamilton equations generated by H.
All Painlevé Hamiltonians have the form
H(t, λ, µ) = f(t, λ)
µ2
2
+ µg(t, λ) + h(t, λ).
For such Hamiltonians it is easy to eliminate µ from the Hamilton equations. One obtains
a second order differential equation for λ of the form
d2λ
dt2
= A(t, λ)
(
dλ
dt
)2
+B(t, λ)
dλ
dt
+ C(t, λ),
A :=
1
2f
∂f
∂λ
, B :=
1
f
∂f
∂t
, C := − g
2
2f
∂f
∂λ
− g
f
∂f
∂t
+ g
∂g
∂λ
+
∂g
∂t
− f ∂h
∂λ
.
From Heun Class Equations to Painlevé Equations 57
B.2 Invariance of Hamilton equations
The Hamilton equations are invariant with respect to various transformations.
� The equations generated by εH(εt, λ, µ) are equivalent to (B.1) and (B.2).
� Let (λ, µ) 7→
(
λ̃, µ̃
)
be a (time-independent) canonical transformation, that means
∂λ̃
∂λ
∂µ̃
∂µ
− ∂λ̃
∂µ
∂µ̃
∂λ
= 1.
Then the Hamilton equations in the new variables
dλ̃
dt
=
∂H
∂µ̃
,
dµ̃
dt
= −∂H
∂λ̃
,
are equivalent to (B.2).
� The Hamilton equations are invariant with respect to the following time-dependent trans-
formation:
λ̃ = t−1λ, µ̃ = tµ, H̃ = H − t−1λ̃µ̃.
B.3 Hamiltonian solvable in quadratures
Let us now consider a Hamiltonian of the form
H(t, λ, µ) = m(t)
(
f(λ)
µ2
2
+ µg(λ) + h(λ)
)
. (B.3)
We will show that it is solvable in quadratures.
First we change the time from t to s, by solving
ds
dt
= m(t).
Using the time s we can replace (B.3) by the time-independent Hamiltonian
H(λ, µ) = f(λ)
µ2
2
+ µg(λ) + h(λ).
Now
E := f(λ)
µ2
2
+ µg(λ) + h(λ)
is a constant of motion, hence we can express µ in terms of λ:
µ =
−g(λ)±
√
g(λ)2 − 2f(λ)(h(λ)− E)
f(λ)
.
We insert this into the first Hamilton equation
dλ
ds
= µf(λ) + g(λ)
obtaining
dλ
ds
= ±
√
g(λ)2 − 2f(λ)(h(λ)− E).
This is clearly solvable in quadratures.
58 J. Dereziński, A. Ishkhanyan and A. Latosiński
Acknowledgements
J.D. and A.I. would like to express their gratitude to Galina Filipuk for very useful discussions
and remarks. A.I. acknowledges the support by the Armenian Science Committee (SC Grant
No. 20RF-171), and the Armenian National Science and Education Fund (ANSEF Grant No. PS-
5701). The work of J.D. and A.L. was supported by National Science Center (Poland) under
the grant UMO-2019/35/B/ST1/01651.
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1 Introduction
2 Second order linear differential equations in the complex domain
2.1 Differential equation and operator
2.2 Singularities of functions
2.3 Singularities of equations
2.4 Sandwiching with functions
2.5 Half-integer rank
2.6 Simplifying the equation
2.7 Solutions in terms of formal power series
2.8 Thomé solutions
2.9 Nonlogarithmic singularities
3 Equations with rational coefficients
3.1 The Mn class and the grounded Mn class
3.2 Generalized Fuchs relation
3.3 Riemann class equations
3.4 Heun class equations
3.5 Deformed Heun class equations
3.6 Classification of Heun class equations
4 From Heun class to Painlevé equations
4.1 Method of isomonodromic deformations
4.2 Isomonodromic deformations in presence of a non-logarithmic singularity
4.3 Isomonodromic deformations of Heun class equations
4.4 Correspondence between Heun class and Painlevé equations
4.5 From Heun (111;1) to Painlevé VI
4.6 From Heun (21;1) to Painlevé V
4.7 From Heun (321;1) to degenerate Painlevé V
4.8 From Heun (2;2) to non-degenerate Painlevé III'
4.9 From Heun (to.2;32)to. to degenerate Painlevé III'
4.10 From Heun (to.32;2)to. to degenerate Painlevé III'
4.11 From Heun (to.32;32)to. to doubly degenerate Painlevé III'
4.12 From Heun (1;3) to Painlevé IV
4.13 From Heun (1;52) to Painlevé 34
4.14 From Heun (;4) to Painlevé II
4.15 From Heun (;72) to Painlevé I
5 Five supertypes of Painlevé equation
5.1 Overview of five supertypes
5.2 Painlevé V or (112)
5.3 Painlevé III' or (22)
5.4 Painlevé IV-34 or (13)
5.5 Painlevé I–II or (4)
A Proof of Theorems 4.1, 4.2 and 4.3
A.1 Preparation for the proof of Theorem 4.1
A.2 First compatibility condition
A.3 Second compatibility condition
A.4 Case A, Part I
A.5 Case A, Part II
A.6 Case B
B Hamilton equations
B.1 From Hamilton equations to second order equations
B.2 Invariance of Hamilton equations
B.3 Hamiltonian solvable in quadratures
References
|
| id | nasplib_isofts_kiev_ua-123456789-211367 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T05:48:35Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Dereziński, Jan Ishkhanyan, Artur Latosiński, Adam 2025-12-30T15:57:32Z 2021 From Heun Class Equations to Painlevé Equations. Jan Dereziński, Artur Ishkhanyan and Adam Latosiński. SIGMA 17 (2021), 056, 59 pages 1815-0659 2020 Mathematics Subject Classification: 34A30; 34B30; 34M55; 34M56 arXiv:2007.05698 https://nasplib.isofts.kiev.ua/handle/123456789/211367 https://doi.org/10.3842/SIGMA.2021.056 In the first part of our paper, we discuss linear 2nd order differential equations in the complex domain, especially Heun class equations, that is, the Heun equation and its confluent cases. The second part of our paper is devoted to Painlevé I-VI equations. Our philosophy is to treat these families of equations in a unified way. This philosophy works especially well for Heun's class equations. We discuss its classification into 5 supertypes, subdivided into 10 types (not counting trivial cases). We also introduce in a unified way deformed Heun class equations, which contain an additional non-logarithmic singularity. We show that there is a direct relationship between deformed Heun class equations and all Painlevé equations. In particular, Painlevé equations can also be divided into 5 supertypes and subdivided into 10 types. This relationship is not so easy to describe in a completely unified way, because the choice of the ''time variable'' may depend on the type. We describe unified treatments for several possible ''time variables''. J.D. and A.I. would like to express their gratitude to Galina Filipuk for very useful discussions and remarks. A.I. acknowledges the support by the Armenian Science Committee (SC Grant No. 20RF-171), and the Armenian National Science and Education Fund (ANSEF Grant No. PS5701). The work of J.D. and A.L. was supported by the National Science Center (Poland) under the grant UMO-2019/35/B/ST1/01651. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications From Heun Class Equations to Painlevé Equations Article published earlier |
| spellingShingle | From Heun Class Equations to Painlevé Equations Dereziński, Jan Ishkhanyan, Artur Latosiński, Adam |
| title | From Heun Class Equations to Painlevé Equations |
| title_full | From Heun Class Equations to Painlevé Equations |
| title_fullStr | From Heun Class Equations to Painlevé Equations |
| title_full_unstemmed | From Heun Class Equations to Painlevé Equations |
| title_short | From Heun Class Equations to Painlevé Equations |
| title_sort | from heun class equations to painlevé equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211367 |
| work_keys_str_mv | AT derezinskijan fromheunclassequationstopainleveequations AT ishkhanyanartur fromheunclassequationstopainleveequations AT latosinskiadam fromheunclassequationstopainleveequations |