Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy
Solutions of the discrete Painlevé II hierarchy are shown to be in relation to a family of Toeplitz determinants describing certain quantities in multicritical random partition models, for which the limiting behavior has been recently considered in the literature. Our proof is based on the Riemann-H...
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| author | Chouteau, Thomas Tarricone, Sofia |
| author_facet | Chouteau, Thomas Tarricone, Sofia |
| citation_txt | Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy. Thomas Chouteau and Sofia Tarricone. SIGMA 19 (2023), 030, 30 pages |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Solutions of the discrete Painlevé II hierarchy are shown to be in relation to a family of Toeplitz determinants describing certain quantities in multicritical random partition models, for which the limiting behavior has been recently considered in the literature. Our proof is based on the Riemann-Hilbert approach for the orthogonal polynomials on the unit circle related to the Toeplitz determinants of interest. This technique allows us to construct a new Lax pair for the discrete Painlevé II hierarchy that is then mapped to the one introduced by Cresswell and Joshi.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 030, 30 pages
Recursion Relation for Toeplitz Determinants
and the Discrete Painlevé II Hierarchy
Thomas CHOUTEAU a and Sofia TARRICONE b
a) Université d’Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers, France
E-mail: thomas.chouteau@univ-angers.fr
b) Institut de Physique Théorique, Université Paris-Saclay, CEA, CNRS,
F-91191 Gif-sur-Yvette, France
E-mail: sofia.tarricone@ipht.fr
URL: https://starricone.netlify.app/
Received December 22, 2022, in final form May 16, 2023; Published online May 28, 2023
https://doi.org/10.3842/SIGMA.2023.030
Abstract. Solutions of the discrete Painlevé II hierarchy are shown to be in relation with
a family of Toeplitz determinants describing certain quantities in multicritical random parti-
tions models, for which the limiting behavior has been recently considered in the literature.
Our proof is based on the Riemann–Hilbert approach for the orthogonal polynomials on
the unit circle related to the Toeplitz determinants of interest. This technique allows us to
construct a new Lax pair for the discrete Painlevé II hierarchy that is then mapped to the
one introduced by Cresswell and Joshi.
Key words: discrete Painlevé equations; orthogonal polynomials; Riemann–Hilbert prob-
lems; Toeplitz determinants
2020 Mathematics Subject Classification: 33E17; 33C47; 35Q15
1 Introduction
Let us consider the symbol φ(z) = ew(z) with
w(z) := v(z) + v
(
z−1
)
and v(z) :=
N∑
j=1
θj
j
zj , (1.1)
for θj being real constants and natural N ≥ 1. The n-th Toeplitz matrix associated to this
symbol and denoted by Tn(φ) is a square (n+1)-dimensional matrix which entries are given by
Tn(φ)i,j := φi−j , i, j = 0, . . . , n. (1.2)
Here for every k ∈ Z, φk is the k-th Fourier coefficient of φ(z), namely
φk =
∫ π
−π
e−ikβφ
(
eiβ
)dβ
2π
,
so that
∑
k∈Z φkz
k = φ(z). Notice that, even though it is not emphasized in our notation, the
functions φk and thus the Toeplitz matrix Tn(φ) explicitly depend on the natural parameter N
which enters in the definition of v(z) in equation (1.1).
This paper is a contribution to the Special Issue on Evolution Equations, Exactly Solvable Mod-
els and Random Matrices in honor of Alexander Its’ 70th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Its.html
mailto:thomas.chouteau@univ-angers.fr
mailto:sofia.tarricone@ipht.fr
https://starricone.netlify.app/
https://doi.org/10.3842/SIGMA.2023.030
https://www.emis.de/journals/SIGMA/Its.html
2 T. Chouteau and S. Tarricone
In the present work, it is indeed the dependence on this parameter N that we want to study.
In particular, we show that the Toeplitz determinants associated to Tn(φ), naturally defined as
DN
n := Dn = det(Tn(φ)), (1.3)
are related to some solutions of a discrete version of the Painlevé II hierarchy, indexed over the
parameter N (the dependence on N is dropped in the rest of the paper). Our interest in these
Toeplitz determinants comes from their appearance in the recent paper [5]. The authors there
consider some probability measures on the set of integer partitions called multicritical Schur
measures, which are a particular case of Schur measures introduced by Okounkov in [23]. They
are generalizations of the classical Poissonized Plancherel measure and they are defined as
P({λ}) = Z−1sλ[θ1, . . . , θN ]2, with Z = exp
(
N∑
i=1
θ2i
i
)
. (1.4)
Here sλ[θ1, . . . , θN ] denotes a Schur symmetric function indexed by a partition λ that can be
expressed as
sλ[θ1, . . . , θN ] = det
i,j
hλi−i+j [θ1, . . . , θN ],
where
∑
k≥0 hkz
k = exp
(∑N
i=1
θi
i z
i
)
. In [5], the authors first used the term multicritical to un-
derline that they obtained a different limiting edge behavior for these Schur measures compared
to the classical case of the Poissonized Plancherel measure (N = 1) which is characterised by
the Tracy–Widom GUE distribution. For more details, we remind to their Theorem 1 or our
discussion in the paragraph “Continuous limit” below, for instance see equation (1.23) where
the higher order Tracy–Widom distributions appear.
In this setting, denoting by λ = (λ1 ≥ λ2 ≥ · · · ≥ 0) a generic integer partition and by
λ′ = (λ′
1 ≥ λ′
2 ≥ · · · ≥ 0) its conjugate partition (namely such that λ′
j = |i : λi ≥ j|), major
quantities of interest of the model are, for any given n ∈ N,
rn := P(λ1 ≤ n) and qn := P(λ′
1 ≤ n), (1.5)
that are often called discrete gap probabilities as random partitions have a natural interpretation
in terms of random configuration of points on the set of semi-integers. Indeed, associating the
set {λi − i + 1/2} ⊂ Z + 1
2 to a partition λ (see [23]), rn and qn can be expressed in terms of
a Fredholm determinant of a discrete kernel which corresponds to the gap probability in the
determinantal point process defined through the same kernel.
According to Geronimo–Case/Borodin–Okounkov formula [7], there is a relation between this
Fredholm determinant and the Toeplitz determinant Dn and this implies that rn and qn (up to
a constant factor) are Toeplitz determinants. It leads to (for instance [5, Propositions 6 and 7]):
qn = e−
∑N
j=1 θ
2
j /jDn−1. (1.6)
For rn instead, one should define θ̃i = (−1)i−1θi and by taking w̃(z) = ṽ(z) + ṽ
(
z−1
)
, where
ṽ(z) is nothing than v(z) with θi replaced by θ̃i as given above, the Toeplitz determinant D̃n
associated to the symbol φ̃(z) = ew̃(z) would give the analogue formula
rn = e−
∑N
j=1 θ̃
2
j /jD̃n−1.
Notice that in the simplest case, when N = 1, the quantities rn and qn coincide. Moreover,
thanks to Schensted’s theorem [27], they are also equal to the discrete probability distribution
Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy 3
function of the length of the longest increasing subsequence of random permutations of size m,
with m distributed as a Poisson random variable.
In the case N = 1, the relation of these quantities with the theory of discrete Painlevé
equations was shown two decades ago independently and through very different methods by
Borodin [6], Baik [2], Adler and van Moerbeke [1] and Forrester and Witte [16].1 In particular,
they all proved that for every n ≥ 1, the following chain of equalities holds
DnDn−2
D2
n−1
=
qn+1qn−1
q2n
=
rn+1rn−1
r2n
= 1− x2n, (1.7)
where xn solves the second order nonlinear difference equation
θ1(xn+1 + xn−1)
(
1− x2n
)
+ nxn = 0, (1.8)
with certain initial conditions. Equation (1.8) is a particular case of the so called discrete
Painlevé II equation [26], a discrete analogue of the classical second order ODE known as the
Painlevé II equation [24]. This means that performing some continuous limit of equation (1.8)
one gets back the Painlevé II equation. The Painlevé II equations, discrete and continuous ones,
depend in general on an additional constant term α ∈ R. In the present work, we consider
the discrete Painlevé II equation and its hierarchy in the homogeneous case where α = 0. Its
continuous limit will correspond as well to the case α = 0.
Remark 1.1. The homogeneous Painlevé II equation admits a famous solution [17], called the
Hastings–McLeod solution, found by requiring a specific boundary condition at ∞. In parallel,
one might wonder what is the large n behavior of the solution xn of the discrete Painlevé II
equation (1.8). Its behavior is expressed in terms of the Bessel functions. First, this is suggested
by the following heuristic arguments. Because of the definition of rn (1.5), as n → ∞, rn tends
to one and according to the equation (1.7), xn tends to zero. Then for large n, the nonlinear
term in equation (1.8) is small compared to the linear ones and the equation (1.8) reduces to
the equation
θ1 (xn+1 + xn−1) + nxn = 0,
which indeed admits J−n(2θ1), the Bessel function of the first kind of order −n, as a solution.
The claim is confirmed by a result of the recent work [9]. The authors there studied the finite
temperature deformation for the discrete Bessel point process. The Fredholm determinant of
the finite temperature discrete Bessel kernel they studied depends on a function σ. In the case
when σ = 1Z′
+
(the characteristic function of the set of positive half integers), the Fredholm
determinant is then equal to rn. Then from [9, equations (1.33) and (1.36) of Theorem III]
together with equation (1.7), one can deduce that for n large x2n ∼ Jn(2θ1)
2 and, because of the
previous discussion, one can conclude
xn ∼ J−n(2θ1) = (−1)nJn(2θ1),
see also Figure 1.
For N > 1, Adler and van Moerbeke presented in [1], a generalization of equation (1.7) by
proving that xn satisfies some recurrence relation written in terms of the Toeplitz lattice Lax
matrices. The main result of our work is a recurrence relation for xn defined via a N -times
iterating discrete operator which establishes the link with the discrete Painlevé II hierarchy [11].
The precise result is stated as below.
1They obtained an analogue of equation (1.7) for Toeplitz determinant associated to symbols which are not
necessarily positive or even real valued.
4 T. Chouteau and S. Tarricone
0 10 20 30 40 50
n
1.00
0.75
0.50
0.25
0.00
0.25
0.50
0.75
1.00 xn
( 1)nJn(2 1)
Figure 1. For N = 1, the graphs of xn and (−1)nJn(2θ1) in function of n for θ1 = 3.
Theorem 1.2. For any fixed N ≥ 1, for the Toeplitz determinants Dn (1.3), n ≥ 1 associated
to the symbol φ(z) (1.1), we have
DnDn−2
D2
n−1
= 1− x2n, (1.9)
where xn solves the 2N order nonlinear difference equation
nxn +
(
−vn − vnPermn + 2xn∆
−1(xn − (∆ + I)xnPermn)
)
LN (0) = 0, (1.10)
where L is a discrete recursion operator defined as
L(un) :=
(
xn+1
(
2∆−1 + I
)
((∆ + I)xnPermn − xn) + vn+1(∆ + I)− xnxn+1
)
un. (1.11)
Here vn := 1− x2n, ∆ denotes the difference operator
∆: un → un+1 − un
and Permn is the transformation of the space C
[
(xj)j∈[[0,2n]]
]
acting by permuting indices in the
following way:
Permn : C
[
(xj)j∈[[0,2n]]
]
−→ C
[
(xj)j∈[[0,2n]]
]
,
P
(
(xn+j)−n⩽j⩽n
)
7−→ P
(
(xn−j)−n⩽j⩽n
)
.
(1.12)
Remark 1.3. According to equation (1.10) and the definition of the operator L (1.11), we
need to perform discrete integrations to compute the N -th equation of the discrete Painlevé II
hierarchy. It is always possible to accomplish this discrete integration. The operator ∆−1,
inverse of the difference operator ∆, is applied to (∆ + I)xnPermn − xn and it is possible to
write this operator as a derivative. Indeed,
(∆ + I)xnPermn − xn = ∆xnPermn + (Permn − I)xn.
The first term on the right hand side is a derivative and because of the definition of Permn, the
second term can be expressed as a derivative.
Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy 5
Equation (1.10), together with the definition of the recursion operator L in (1.11), of the
quantity vn and of the transformation Permn in (1.12) is indeed the N -th member of the discrete
Painlevé II hierarchy. The first equations of the hierarchy read as
N = 1: nxn + θ1(xn+1 + xn−1)
(
1− x2n
)
= 0, (1.13)
N = 2: nxn + θ1
(
1− x2n
)
(xn+1 + xn−1) + θ2
(
1− x2n
)
×
(
xn+2
(
1− x2n+1
)
+ xn−2
(
1− x2n−1
)
− xn(xn+1 + xn−1)
2
)
= 0, (1.14)
N = 3: nxn + θ1
(
1− x2n
)
(xn+1 + xn−1) + θ2
(
1− x2n
)(
xn+2
(
1− x2n+1
)
+ xn−2
(
1− x2n−1
)
− xn(xn+1 + xn−1)
2
)
+ θ3
(
1− x2n
)(
x2n(xn+1 + xn−1)
3
+ xn+3
(
1− x2n+2
)(
1− x2n+1
)
+ xn−3
(
1− x2n−2
)(
1− x2n−1
))
+ θ3
(
1− x2n
)(
−2xn(xn+1 + xn−1)
(
xn+2
(
1− x2n+1
)
+ xn−2
(
1− x2n−1
))
− xn−1x
2
n−2
(
1− x2n−1
))
+ θ3
(
1− x2n
)(
−xn+1x
2
n+2
(
1− x2n+1
)
− xn+1xn−1(xn+1 + xn−1)
)
= 0 (1.15)
with the first one coinciding with the discrete Painlevé II equation (1.8). Computations with
the operator (1.11) introduced in Theorem 1.2 for N = 1 and 2 are done in Example 3.11.
Remark 1.4. The same heuristic argument used in Remark 1.1 applies also when N > 1
(since xn still tends to zero as n → ∞), thus suggesting that the N -th equation of the discrete
Painlevé II hierarchy reduces to a linear discrete equation for large n. For N = 2 and 3, the
reduced equations are
N = 2: nxn + θ1(xn+1 + xn−1) + θ2(xn+2 + xn−2) = 0,
N = 3: nxn + θ1(xn+1 + xn−1) + θ2(xn+2 + xn−2) + θ3(xn+3 + xn−3) = 0.
Similar recurrence relations appeared in [12] for the multivariable generalized Bessel functions
(GBFs). These generalized Bessel functions were discussed in [21, 23] in the context of Schur
measures for random partitions and generalizations of the previous recurrence equations were
introduced (in particular, see in [21, equation (3.2b)]). We denote by J
(N)
n (ξ1, . . . , ξN ) a N -
variable GBFs of order n. In [12], J
(N)
n (ξ1, . . . , ξN ) is defined as a discrete convolution product
of N Bessel functions. In particular, if j
(k)
n (ξ) is the n-th Fourier coefficient of the function
β → e2iξ sin(kβ) then
J (N)
n (ξ1, . . . , ξN ) := j(N)
n (ξN ) ∗ j(N−1)
n (ξN−1) ∗ · · · ∗ j(1)n (ξ1)(n),
where ∗ denotes the discrete convolution.
In the case N = 1, the symbol we considered was φ
(
eiβ
)
= eθ1(e
iβ+e−iβ) = e2θ1 cos(β) and the
large n asymptotic behavior of xn was given by J−n(2θ1) which is the n-th Fourier coefficient of
the function β → eθ1(e
iβ−e−iβ) up to a constant (−1)n.
For N > 1, the symbol is φN (eiβ) =
∏N
k=1 e
θk
k
(eikβ+e−ikβ) =
∏N
k=1 e
2
θk
k
cos(kβ). Then, we
conjecture that the large n asymptotic behavior of x
(N)
n would be given by the n-th Fourier
coefficient of β →
∏N
k=1 e
(−1)k+1θk
k
(eikβ−e−ikβ) which is precisely J
(N)
n (ξ1, . . . , ξN ) up to some
constant and proper rescaling:
x(N)
n ∼ (−1)nJ (N)
n
((
(−1)i
2
i
θi
)
1⩽i⩽N
)
,
see also Figure 2.
6 T. Chouteau and S. Tarricone
0 10 20 30 40 50
n
1.00
0.75
0.50
0.25
0.00
0.25
0.50
0.75
1.00 x(2)
n
( 1)nJ(2)
n (2 1, 2)
0 10 20 30 40 50
n
1.00
0.75
0.50
0.25
0.00
0.25
0.50
0.75
1.00 x(3)
n
( 1)nJ(3)
n (2 1, 2, 2/3 3)
Figure 2. The graphs of x
(N)
n and (−1)nJ
(N)
n
(
(θi)1⩽i⩽N
)
(for N = 2 on left and N = 3 on the right) in
function of n for θ1 = 3, θ2 = 1.2 and θ3 = 2.6.
Remark 1.5. Notice that for N = 1, 2 the equations (1.13) and (1.14) coincide with the ones
found in [1]. Also notice that in the physics literature, Periwal and Schewitz [25] found similar
discrete equations for N = 1, 2 (with different coefficients sign) in the context of unitary matrix
models and used their solutions to evaluate the behavior of some typical integrals in the large-
dimensional limit passing through the continuous limit of their discrete equations. For N = 1,
the discrete Painlevé II equation was also found in [18] as a particular case of the string equation
for the full unitary matrix model, i.e., for w(z) = θ1z + θ−1z
−1. The dependence in θ±1 of xn
(and some other x∗n) was also studied there and it produced some evolution equations related,
after some change of variables, to the two-dimensional Toda equations. This would suggest that
for the general case N > 1, the dependence of xn on times θ1, . . . , θN would be related to the
one-dimensional Toda hierarchy (see also [23]).
The first construction of a discrete Painlevé II hierarchy in [11] used the integrability property
of the continuous one, in the following sense. It is well known that the classical Painlevé II equa-
tion admits an entire hierarchy of higher order analogues. Indeed, this equation can be obtained
as a self-similarity reduction of the modified KdV equation. Thus, the higher order members of
the Painlevé II hierarchy are but analogue self-similarity reductions of the corresponding higher
order members of the modified KdV hierarchy (see, e.g., [14]). In some way, this implies that
the Lax representation of the KdV hierarchy in terms of isospectral deformations becomes for
the Painlevé II hierarchy a Lax representation in terms of isomonodromic deformations [10].
In [11] then, the discrete Painlevé II hierarchy is defined as the compatibility condition of
a sort of “discretization” of the Lax representation of the Painlevé II hierarchy. In particular,
they considered the compatibility condition of a linear 2×2 matrix-valued system of the following
type:
Φn+1(z) = Ln(z)Φn(z),
∂
∂z
Φn(z) = Mn(z)Φn(z), (1.16)
where the coefficients Ln(z), Mn(z) are explicit matrix-valued rational function in z, depending
on xℓ, ℓ = n+N, . . . , n−N , in some recursive (on N) way. This allows the authors there to com-
Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy 7
pactly write the N -th discrete Painlevé II equation using some recursion operators. The linear
system that we obtain in Proposition 2.11 and that encodes our hierarchy as written in (1.10) is
mapped into the one of [11] through an explicit transformation, as shown in Proposition 2.18,
thus implying that (1.10) is indeed the same discrete Painlevé II hierarchy.
Continuous limit. The aim of this paragraph is to explain heuristically the reason why our
result given in Theorem 1.2 can be considered as the discrete analogue of the generalized Tracy–
Widom formula for higher order Airy kernels (namely, the result contained in [8, Theorem 1.1],
case τi = 0).
For N = 1, Borodin in [6] already pointed out that formula (1.7) with (1.8) can be seen as
a discrete analogue of the classical Tracy–Widom formula for the GUE Tracy–Widom distribu-
tion [28, 29]. In other words, he described how to pass from the left to the right in the picture
below:
Discrete case
DnDn−2 −D2
n−1
D2
n−1
= −x2n
with nxn + θ
(
1− x2n
)
(xn+1 + xn−1) = 0,
Continuous case
d2
dt2
log det(1−KAi|(t,+∞)) = −u2(t)
with u′′(t) = 2u3(t) + tu(t),
u(t) ∼
t→∞
Ai(t),
Baik–Deift–Johansson
where Ai(t) denotes the classical Airy function and KAi denotes the integral operator acting
on L2(R) through the Airy kernel. This connection was achieved by using the scaling limit
computed by Baik, Deift and Johansonn in [3] for the distribution of the first part of partitions
in the Poissonized Plancherel random partition model (which is recovered in [5, Theorem 1] for
N = 1). In some way, as emphasized by Borodin, their result not only assures the existence of
a limiting function for the Dn, in this case D(t) = det(1−KAi|(t,+∞)), for a certain continuous
variable t. It also encodes already how the discrete function xn, should be rescaled in terms of
a differentiable function u(t) to get back, from the recursion relation for Dn, the Tracy–Widom
formula.
To generalize this result for the case N > 1, we proceed by adapting the method used
by Borodin in [6] for N = 1 to the higher order cases, using the scaling proposed in [5]2 for
the multicritical case (notice that their n corresponds to our N), instead of the Baik–Deift–
Johansson’s one that only holds for N = 1.
We recall that Dn is the Toeplitz determinant associated to the symbol φ(z) (1.1) (which
depends on θi, i = 1, . . . , N and thus on N). In the following discussion, we write explicitly
the dependence on the family of parameters (θi), i = 1, . . . , N of Dn = Dn(θi), xn = xn(θi),
rn = rn(θi) and qn = qn(θi). Consider equation (1.9) written in terms of the Toeplitz determi-
nants Dn(θi) in this way
Dn−2(θi)Dn(θi)−D2
n−1(θi)
D2
n−1(θi)
= −x2n(θi). (1.17)
From the equation (1.6), this previous equation can be expressed in terms of qn(θi) defined
as (1.5). It becomes
qn−1(θi)qn+1(θi)− q2n(θi)
q2n(θi)
= −x2n(θi). (1.18)
2Up to the correction of the typo d → d−1 in their statement of Theorem 1.
8 T. Chouteau and S. Tarricone
According to [5, Lemma 8], with the change of parameters θ̃i = (−1)i−1θi, we have qn(θi) =
rn
(
θ̃i
)
. Thus equation (1.18) now reads as
rn−1
(
θ̃i
)
rn+1
(
θ̃i
)
− r2n
(
θ̃i
)
r2n
(
θ̃i
) = −x2n(θi). (1.19)
Following the scaling limit described in [5, Theorem 1], we define the following scaling for the
discrete variable n:
n = bθ + tθ
1
2N+1d−
1
2N+1 ⇐⇒ t = (n− bθ)θ−
1
2N+1d
1
2N+1 (1.20)
with b, d defined as
b =
N + 1
N
, d =
(
2N
N − 1
)
and choose θ̃i (resp. θi) all proportional to θ = θ̃1 = θ1 in the following way:
θ̃i = (−1)i−1 (N − 1)!(N + 1)!
(N − i)!(N + i)!
θ, i = 1, . . . , N,
respectively,
θi =
(N − 1)!(N + 1)!
(N − i)!(N + i)!
θ, i = 1, . . . , N. (1.21)
Now recall the definition of rn
(
θ̃i
)
(1.5) in function of P = Pθ̃i
(see equation (1.4) for the
definition of P and the dependence on the family of parameters (θi)). From the previous scaling,
it is now possible to express rn
(
θ̃i
)
in function of t and θ
rn
(
θ̃i
)
= Pθ̃i
(
λ1 − bθ
(θd−1)
1
2N+1
⩽ t
)
(1.22)
and according to [5, Theorem 1], the limiting behavior of the probability distribution function
of λ1 in this setting is given by
lim
θ→+∞
rn
(
θ̃i
)
= lim
θ→+∞
Pθ̃i
(
λ1 − bθ(
θd−1
) 1
2N+1
⩽ t
)
= FN (t),
FN (t) = det(1−KAi2N+1
|(t,∞)), (1.23)
where KAi2N+1
is the integral operator acting with higher order Airy kernel (see, for instance,
in [5, equation (2.7)]).
As we did for rn
(
θ̃i
)
in equation (1.22), we express rn+1
(
θ̃i
)
and rn−1
(
θ̃i
)
in function of t
and θ:
rn±1
(
θ̃i
)
= Pθ̃i
(
λ1 − bθ(
θd−1
) 1
2N+1
⩽ t±
(
θd−1
)− 1
2N+1
)
.
With this discussion and this scaling for n, (θi) and
(
θ̃i
)
, we deduce that
− lim
θ→+∞
x2n(θi)(
θd−1
)− 2
2N+1
= lim
θ→+∞
rn−1
(
θ̃i
)
rn+1
(
θ̃i
)
− r2n
(
θ̃i
)(
θd−1
)− 2
2N+1 r2n
(
θ̃i
) =
d2
dt2
logFN (t),
where the first equality comes from equation (1.19) and the second from equation (1.23).
Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy 9
From now on, we drop the dependence on θi, i = 1, . . . , N in the notation. The previous
equation suggests that, in order to be consistent with [8, Theorem 1.1], the discrete function xn
appearing in formula (1.17) in the scaling (1.20) for n and (1.21) for (θi) limit should be
−x2n ∼ −(θ)−
2
2N+1d
2
2N+1u2(t)
with u(t) solution of the N -th equation of the Painlevé II hierarchy. This can be proved directly
by computing the scaling limit of the equations of the discrete Painlevé II hierarchy we found
for xn in Theorem 1.2. Indeed, for every fixed N , we write xn as
xn = (−1)nθ−
1
2N+1d
1
2N+1u(t) (1.24)
with u(t) a smooth function of the variable t defined as in equation (1.20). Now recall that xn
solves the discrete equation (1.10) of order 2N for every N ≥ 1. The continuous limit of the
discrete equations of the hierarchy (1.10), under the definition of xn (1.24) and the scaling of the
parameters θi as (1.21), gives the equations of the classical Painlevé II hierarchy. For any fixed N
the computation should be done in the same way: consider the N -th discrete equation of the
hierarchy (1.10) and replace each θi with the values given in formula (1.21). Then substitute xn
with the definition in (1.24) and for θ → +∞ compute the asymptotic expansion of every
term xn+K ∝ u
(
t +Kθ−
1
2N+1d
1
2N+1
)
, K = −N, . . . , N appearing in the discrete equation. The
coefficient of θ−1 resulting after this procedure coincides indeed with the N -th equation of the
Painlevé II hierarchy. For N = 1, 2, 3, the computations are explicitly done in the Appendix A.
Remark 1.6. It is worthy to mention that in [8], the authors also consider a generalization of
the Fredholm determinant FN (t), recalled here in (1.23), depending on additional parameters τi.
Those are related to solutions of the general Painlevé II hierarchy, which depends as well on
the τi. With the scaling as in [5] for the θi’s, the continuous limit for our discrete equations leads
to the Painlevé II hierarchy with τi = 0 for all i. This is consistent with the fact that the limiting
behavior in [5], written here in equation (1.23), involves indeed the Fredholm determinant FN (t)
corresponding to τi = 0 for all i (the same already appeared in [22]).
Methodology and outline. The rest of the work is devoted to prove Theorem 1.2. In order
to do so, we introduce the classical Riemann–Hilbert characterization [4] of the family of orthog-
onal polynomials on the unit circle (OPUC for brevity) with respect to a measure defined by
the symbol φ(z). Classical results from orthogonal polynomials theory allow to achieve almost
directly formula (1.17) where xn is defined as the constant term of the n-th monic orthogonal
polynomial of the family. The Riemann–Hilbert problem for the OPUC is then used to deduce
a linear system of the same type of (1.16) which is proven to be in relation with the Lax pair
introduced by Cresswell and Joshi [11] for the discrete Painlevé II hierarchy. This is done in Sec-
tion 2. The explicit computation of the Lax pair together with the construction of the recursion
operator and the hierarchy for xn as written in (1.10) are done in Section 3.
2 OPUC: the Riemann–Hilbert approach and a discrete
Painlevé II Lax pair
In this section, we introduce the relevant family of orthogonal polynomials on the unit circle.
We recall some of their properties and their Riemann–Hilbert characterization. Afterward we
derive a Lax pair associated to the Riemann–Hilbert problem and establish the relation with
the Lax pair for discrete Painlevé II hierarchy (1.16) introduced by Cresswell and Joshi [11].
The proofs of the results for orthogonal polynomials stated in here can be found in the classical
reference [4].
10 T. Chouteau and S. Tarricone
We denote by S1 the unit circle in C counterclockwise oriented. We consider the following
positive measure on S1 (absolutely continuous w.r.t. the Lebesgue measure there):
dµ(β) =
ew(eiβ)
2π
dβ, (2.1)
where the function w(z) for any z ∈ C is given as in equation (1.1). The family of orthogonal
polynomials on the unit circle (OPUC) w.r.t. the measure (2.1) is defined as the collection of
polynomials {pn(z)}n∈N written as
pn(z) = κnz
n + · · ·+ κ0, κn > 0 (2.2)
and such that the following relation holds for any indices k, h∫ π
−π
pk
(
eiβ
)
ph
(
eiβ
)dµ(β)
2π
= δk,h.
The family of monic orthogonal polynomials {πn(z)} associated to the previous ones is defined
in analogue way, so that pn(z) = κnπn(z).
2.1 Toeplitz determinants related to OPUC
We recall that φ(z) = ew(z), z ∈ S1 with w(z) defined as in (1.1) and that we defined Dn :=
det(Tn(φ)) (by conventionD−1 = 1) to be the n-Toeplitz determinant associated to the symbol φ
(see equations (1.2) and (1.3)). Because φ(z) is a real nonnegative function, Dn ∈ R>0.
Proposition 2.1. If φ(z) is a real nonnegative function, we have that
pℓ(z) =
1√
DℓDℓ−1
det
φ0 φ−1 . . . φ−ℓ+1 φ−ℓ
φ1 φ0 . . . φ−ℓ+2 φ−ℓ+1
...
...
. . .
...
...
φℓ−1 φℓ−2 . . . φ0 φ−1
1 z . . . zℓ−1 zℓ
, ℓ ≥ 0. (2.3)
Proof. The proof is similar to the one for the orthogonal polynomials on the real line, that can
be found, e.g., [13, equation (3.5)], and following discussion. ■
Corollary 2.2. The ratio of two consecutive Toeplitz determinants is expressed as
Dℓ−1
Dℓ
= κ2ℓ , ℓ ≥ 0. (2.4)
Proof. Thanks to formula (2.3), we have that
pℓ(z) =
1√
DℓDℓ−1
det
φ0 φ−1 . . . φ−ℓ+1
φ1 φ0 . . . φ−ℓ+2
...
...
. . .
...
φℓ−1 φℓ−2 . . . φ0
zℓ + · · · =
√
Dℓ−1
Dℓ
zℓ + · · · ,
and by definition pℓ(z) = κℓπℓ(z) with the latter being the ℓ-th monic orthogonal polynomial
on S1. Thus formula (2.4) follows. ■
Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy 11
2.2 Riemann–Hilbert problem associated to OPUC
The family {πn} of orthogonal polynomials has a well-known characterization in terms of a 2×2
dimensional Riemann–Hilbert problem, also depending on n ≥ 0.
Riemann–Hilbert Problem 2.3. The function Y (z) := Y (n, θj ; z) : C → GL(2,C) has the
following properties:
(1) Y (z) is analytic for every z ∈ C \ S1;
(2) Y (z) has continuous boundary values Y±(z) while approaching non-tangentially S1 either
from the left or from the right, and they are related for all z ∈ S1 through
Y+(z) = Y−(z)JY (z), with JY (z) =
(
1 z−new(z)
0 1
)
;
(3) Y (z) is normalized at ∞ as
Y (z) ∼
(
I +
∞∑
j=1
Yj(n, θj)
zj
)
znσ3 , z → ∞,
where σ3 denotes the Pauli’s matrix σ3 :=
(
1 0
0 −1
)
.
It is known from [3] that the above Riemann–Hilbert problem, for each n ≥ 0, admits a unique
solution which is explicitly written in terms of the family {πn(z)}. Before stating the result, we
introduce the following notation. For every polynomial q(z), z ∈ C, its reverse polynomial q∗(z)
is defined as the polynomial of the same degree such that
q∗(z) := znq
(
z̄−1
)
.
For every
(
Lp
(
S1
))
function f(y), its Cauchy transform Cf(z) is defined for any z /∈ S1 as
(Cf(y)) (z) := 1
2πi
∫
S1
f(y)
y − z
dy.
Remark 2.4. Notice that the results in [3] for the Riemann–Hilbert characterization a family of
orthogonal polynomials on the unit circle are a sort of extension of the results known from [15, 20]
for the case of orthogonal polynomials on the real line.
Theorem 2.5. For every n ≥ 0, the Riemann–Hilbert Problem 2.3 admits a unique solution Y (z)
that is written as
Y (z) =
(
πn(z) C
(
y−nπn(y)e
w(y)
)
(z)
−κ2n−1π
∗
n−1(z) −κ2n−1C
(
y−nπ∗
n−1(y)e
w(y)
)
(z)
)
. (2.5)
Moreover, det(Y (z)) ≡ 1.
Proof. See [3, Lemma 4.1]. ■
The solution Y (z) has a symmetry which will be very useful in the following section.
Corollary 2.6. The unique solution Y (z) of the Riemann–Hilbert Problem 2.3 is such that
Y (z) = σ3Y (0)−1Y
(
z−1
)
znσ3σ3, (2.6)
Y (z) = Y (z̄). (2.7)
12 T. Chouteau and S. Tarricone
Proof. See [4, Proposition 5.12]. ■
Notice that the factor Y (0) = Y (n, θj ; 0) appearing in equation (2.6) has a very explicit form
by equation (2.5). This will be useful in the following sections.
Lemma 2.7. For every n ≥ 0, we have
Y (0) = Y (n, θj ; 0) =
(
xn κ−2
n
−κ2n−1 xn
)
, (2.8)
where we denoted with xn := πn(0), and κn is defined as in equation (2.2). Moreover, we have
κ2n−1
κ2n
= 1− x2n, (2.9)
and we have xn ∈ R.
Proof. The first column of Y (n; 0) directly follows from the evaluation in z = 0 of Y (n; z)
as given in equation (2.5). Indeed, Y 11(n; 0) = πn(0) and Y 21(n; 0) = −κ2n−1π
∗
n−1(0) but we
observe that
π∗
n−1(0) = zn−1πn−1
(
z̄−1
)∣∣
z=0
= zn−1
(
z−(n−1) + · · ·+ πn−1(0)
)∣∣
z=0
= 1.
Thus we conclude that Y 21(n; 0) = −κ2n−1. For what concerns the second column of Y (n; 0),
we first find the (2, 2)-entry. This is indeed easily deduced from the symmetry given in (2.6).
In the limit for z → ∞ it gives
Y (n; 0) = σ3Y
−1(n; 0)σ3,
thus Y 22(n; 0) = Y 11(n; 0) = πn(0). Finally, for the entry (1, 2) of Y (n; 0), we compute it
explicitly using the orthonormality property of the polynomials pm(z)
Y 12(n; 0) =
1
2πi
∫
S1
πn(s)s
−nw(s)
s
ds =
∫ π
−π
πn
(
eiθ
)
einθw
(
eiθ
)dθ
2π
=
1
κ2n
∫ π
−π
pn
(
eiθ
)
pn
(
eiθ
)
w
(
eiθ
)dθ
2π
=
1
κ2n
.
Equation (2.9) comes from the fact that det(Y (n, θj ; z)) = 1 identically in z and so in particular
for z = 0 by writing Y (n, θj ; 0) as in equation (2.8), relation (2.9) is obtained.
Finally, the fact that xn is real follows from the entry (1, 1) of equation (2.7) together with
equation (2.5). ■
At this point, we are already able to express the ratio of Toeplitz determinants in terms of
the constant term of the monic orthogonal polynomials, as follows.
Corollary 2.8. For every n ≥ 1, the Toeplitz determinants Dn satisfy the recursion relation
Dn−2Dn
D2
n−1
= 1− x2n. (2.10)
Proof. Putting together equation (2.9) with equation (2.4) (for two consecutive integers) we
obtain the recursion relation (2.10). ■
Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy 13
We emphasize again that the symbol φ(z) actually depends on the natural parameter N , so
the Toeplitz determinants Dn, n ≥ 1 (1.3) do as well as xn = πn(0), n ≥ 1 do (since it is the
constant coefficient of the n-th monic OPUC w.r.t. the N -depending measure (2.1), (1.1)). The
N -dependence of the latter will be emphasized in the following section, where xn is proved to
be a solution of the N -th higher order generalization of the discrete Painlevé II equation.
We consider now the following matrix-valued function
Ψ(n, θj ; z) :=
(
1 0
0 κ−2
n
)
Y (n, θj ; z)
(
1 0
0 zn
)
ew(z)
σ3
2 . (2.11)
Thanks to the properties of Y (z;n, θj) from the Riemann–Hilbert Problem 2.3 one can prove
that Ψ(n, θj ; z) satisfies the following Riemann–Hilbert problem.
Riemann–Hilbert Problem 2.9. The function Ψ(z) := Ψ(n, θj ; z) : C → GL(2,C) has the
following properties:
(1) Ψ(z) is analytic for every z ∈ C \
{
S1 ∪ {0}
}
;
(2) Ψ(z) has continuous boundary values Ψ±(z) while approaching non-tangentially S1 either
from the left or from the right, and they are related for all z ∈ S1 through
Ψ+(z) = Ψ−(z)J0, J0 =
(
1 1
0 1
)
; (2.12)
(3) Ψ(z) has asymptotic behavior near 0 given by
Ψ(z) ∼
(
1 0
0 κ−2
n
)
Y (0)
(
I +
∞∑
j=1
zj Ỹj(n)
)(
1 0
0 zn
)
ew(z)
σ3
2 , z → 0; (2.13)
(4) Ψ(z) has asymptotic behavior near ∞ given by
Ψ(z) ∼
(
1 0
0 κ−2
n
)(
I +
∞∑
j=1
Yj(n)
zj
)(
zn 0
0 1
)
ew(z)
σ3
2 , |z| → ∞. (2.14)
Proposition 2.10. The function Ψ(n, θj ; z) defined in (2.11) solves the Riemann–Hilbert Prob-
lem 2.9.
Proof. The analyticity condition and the asymptotic expansions at 0, ∞ given in (2.13), (2.14)
follows directly from the definition (2.11) and the fact that Y (z) solves the Riemann–Hilbert
Problem 2.3. Condition (2.12) follows from direct computation
Ψ(z)+ =
(
1 0
0 κ−2
n
)
Y+(z)
(
1 0
0 zn
)
ew(z)
σ3
2 =
(
1 0
0 κ−2
n
)
Y−(z)JY (z)
(
1 0
0 zn
)
ew(z)
σ3
2
= Ψ−(z)
(
1 0
0 z−n
)
e−w(z)
σ3
2
(
1 z−new(z)
0 1
)(
1 0
0 zn
)
ew(z)
σ3
2 = Ψ−(z)
(
1 1
0 1
)
. ■
2.3 A linear differential system for Ψ(z)
From the solution of the Riemann–Hilbert Problem 2.9, we deduce the following equations (in the
following we omit in Ψ the dependence on θj that should be considered only as parameters and
not actual variables like n, z).
14 T. Chouteau and S. Tarricone
Proposition 2.11. We have
Ψ(n+ 1; z) = U(n; z)Ψ(n; z), ∂zΨ(n; z) = T (n; z)Ψ(n; z) (2.15)
with
U(n; z) :=
(
z + xnxn+1 −xn+1
−
(
1− x2n+1
)
xn 1− x2n+1
)
= σ+z + U0(n), (2.16)
where σ+ :=
(
1 0
0 0
)
and
T (n; z) := T1(n)z
N−1 + T2(n)z
N−2 + · · ·+ T2N+1(n)z
−N−1 =
2N+1∑
k=1
Tkz
N−k, (2.17)
where
T1(n) =
θN
2
σ3. (2.18)
Remark 2.12. The coefficient (Ti(n))2⩽i⩽2N+1 defined in equation (2.17) will be computed in
Section 3.
Proof. We first prove the first equation. We start by defining the quantity
U(n; z) := Ψ(n+ 1; z)Ψ−1(n; z).
Since the jump condition for Ψ(z) (2.12) is independent of n, U(n; z) is analytic everywhere.
Plugging in equation (2.14), we have the expansion at ∞
U(n; z) =
(
1 0
0 κ−2
n+1
)(
I +
Y1(n+ 1)
z
+O
(
z−2
))
z(n+1)σ3
(
1 0
0 z
)
z−nσ3
×
(
I − Y1(n)
z
+O
(
z−2
))(1 0
0 κ2n
)
,
from which we deduce that U(n; z) is a polynomial in z of degree 1, by Liouville theorem.
Moreover, its matrix-valued coefficient are written as
U(n; z) = z
(
1 0
0 0
)
+
(
1 0
0 κ−2
n+1
)
Y (n+ 1; 0)
(
1 0
0 0
)
Y −1(n; 0)
(
1 0
0 κ2n
)
=U0(n)
.
Doing the computation and using equation (2.8), we obtain
U0(n) =
(
Y 11(n+ 1; 0)Y 22(n; 0) −κ2nY
11(n+ 1; 0)Y 12(n; 0)
κ−2
n+1Y
21(n+ 1; 0)Y 22(n, 0) −Y 21(n+ 1; 0)Y 12(n; 0)
)
=
(
xn+1xn −xn+1
−
(
1− x2n+1
)
xn 1− x2n+1
)
.
For what concerns the second equation, we define T (n; z) := ∂zΨ(n; z)Ψ−1(n; z). From the
asymptotic behavior of Ψ(n; z) at 0 and ∞, we can deduce that T (n; z) is a meromorphic
function in z with behavior at ∞ described by
T (n; z) ∼
(
1 0
0 κ−2
n
)(
I +
Y1(n)
z
+O
(
z−2
))V ′(z)
2
σ3
(
I − Y1(n)
z
+O
(
z−2
))(1 0
0 κ2n
)
Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy 15
(polynomial behavior of degree N − 1) while at 0 its behavior is described by
T (n; z) ∼
(
1 0
0 κ−2
n
)
Y (n, 0)
(
I + Ỹ1(n)z +O
(
z2
))
× −V ′(z−1)
2z2
σ3
(
I − Ỹ1(n)z +O
(
z2
))(1 0
0 κ2n
)
,
i.e., there is a pole of order N + 1. In conclusion, we can write
T (n; z) =
θN
2
σ3z
N−1 + T2(n)z
N−2 + · · ·+ T2N+1(n)z
−N−1. ■
Moreover, thanks to the symmetry for the solution of the Riemann–Hilbert problem Y (z)
stated in (2.6), we have that the coefficient matrix T (n; z) satisfies a symmetry property.
Proposition 2.13. T (n; z) has the following symmetry:
T (n; z−1) = −z2
(
K(n)T (n; z)K(n)−1 − nz−1I2
)
(2.19)
with K(n) :=
(
1 0
0 κ−2
n
)
Y (n; 0)σ3
(
1 0
0 κ2n
)
.
Remark 2.14. Notice that for all n, the matrix K(n) is s.t. K(n)−1 = K(n) since we have the
identity x2n +
κ2
n−1
κ2
n
= 1.
Proof. On the one hand,
∂z
(
Ψ
(
n; z−1
))
= − 1
z2
T
(
n; z−1
)
Ψ
(
n; z−1
)
.
On the other hand, using the symmetry (2.6) for Y we deduce the following symmetry for Ψ:
Ψ
(
n; z−1
)
= z−n
(
1 0
0 κ−2
n
)
Y (0)σ3
(
1 0
0 κ2n
)
Ψ(n; z)σ3.
This previous equation leads to
∂z
(
Ψ
(
n; z−1
))
= z−n
(
1 0
0 κ−2
n
)
Y (0)σ3
(
1 0
0 κ2n
)
∂zΨ(n; z)σ3 − nz−1Ψ
(
n; z−1
)
.
Then
T
(
n; z−1
)
= − z2
((
1 0
0 κ−2
n
)
Y (0)σ3
(
1 0
0 κ2n
)
T (n; z)
(
1 0
0 κ−2
n
)
σ3Y (0)−1
×
(
1 0
0 κ2n
)
− nz−1I2
)
. ■
The symmetry (2.19) reflects on the coefficients Tk(n), k = 1, . . . , 2N + 1 as written below.
Corollary 2.15. The coefficients Tk(n), k = 1, . . . , 2N + 1 satisfy
Tj(n) = −K(n)T2N+2−j(n)K(n)−1, j = 1, . . . , N, (2.20)
TN+1(n) = −K(n)TN+1(n)K(n)−1 + nI2. (2.21)
16 T. Chouteau and S. Tarricone
Proof. Indeed, by replacing the exact shape of T (n; z) in equation (2.19), we have
2N+1∑
k=1
Tk(n)z
−N+k = T
(
n; z−1
)
= −z2
(
2N+1∑
k=1
KTk(n)K
−1zN−k − nz−1I2
)
= −
2N+1∑
k=1
KTk(n)K
−1zN+2−k + nzI2
= −
2N+1∑
j=1
KT2N+2−j(n)K
−1z−N+j + nzI2,
so looking at the powers z−N+j for j = 1, . . . , N , we get equation (2.20) and for j = N + 1, we
get equation (2.21). ■
Notice first that from equations (2.20) if the first N + 1 coefficients of T (n; z) are known,
then we can obtain the remaining ones. Second, notice that the coefficient TN+1(n) plays an
important role since it solves an equation, the one given in (2.21).
2.4 Relation with the Cresswell–Joshi Lax pair
To conclude this section, we describe how the Lax pair (2.15) is related with the one of the
discrete Painlevé II hierarchy (1.16) originally introduced by Cresswell and Joshi in [11] as
follows.
Definition 2.16. A Lax pair for the discrete Painlevé II hierarchy is given by a pair of matrices
(Ln(z),Mn(z)), defining the coefficients of a discrete-differential system for a matrix-valued
function Φ(n; z), such as
Φ(n+ 1; z) =
(
z xn
xn 1/z
)
Φ(n; z) = Ln(z)Φ(n; z), (2.22)
∂
∂z
Φ(n; z) = Mn(z)Φ(n; z), (2.23)
with the property that
Mn(z) =
(
An(z) Bn(z)
Cn(z) −An(z)
)
with An, Bn and Cn are rational in z (and depending also on N).
Remark 2.17. Specifically, in [11, Section 3.1], the authors proved that the compatibility
condition of the system of equations (2.22) and (2.23) defines the coefficients of the matrixMn(z),
leaving in turns only one discrete equation of order 2N for xn. This is defined as the N -th
member of the discrete Painlevé II hierarchy.
We establish now a link between this Lax Pair and the system (2.15) we obtained starting
from the OPUC. We define
Φ(n; z) := σ3
(
z−n+3/2 0
0 z−n+1/2
)(
1 0
−xn−1 1
)
Ψ
(
n− 1; z2
)
.
Proposition 2.18. Φ(n; z) defined as above satisfies the system of equations (2.22) and (2.23).
Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy 17
Proof. First we compute the discrete equation for Φ(n; z). From the definition, we have
Φ(n+ 1; z) = σ3
(
z−n+1/2 0
0 z−n−1/2
)(
1 0
−xn 1
)
Ψ
(
n; z2
)
.
According to equation (2.15),
Φ(n+ 1; z) = σ3
(
z−n+1/2 0
0 z−n−1/2
)(
1 0
−xn 1
)
U
(
n− 1; z2
)
Ψ
(
n− 1; z2
)
= σ3
(
z−n+1/2 0
0 z−n−1/2
)(
1 0
−xn 1
)
U
(
n− 1; z2
)( 1 0
xn−1 1
)
×
(
zn−3/2 0
0 zn−1/2
)
σ3Φ(n; z) =
(
z xn
xn 1/z
)
Φ(n; z).
Now we compute the derivative with respect to z.
Defining Mn(z) :=
(
∂
∂zΦ(n; z)
)
Φ(n; z)−1, similar computations lead to
Mn(z) = z−1σ3
(
−n+ 3/2 0
0 −n+ 1/2
)
σ3 + 2zσ3
(
z 0
0 1
)(
1 0
−xn−1 1
)
× T
(
n− 1; z2
)( 1 0
xn−1 1
)(
z−1 0
0 1
)
σ3. (2.24)
We need to prove two things: first the trace of Mn(z) is null and then entries of Mn(z) are
rational in z.
For the trace of Mn(z) we use the fact that Tr(T (n; z)) = nz−1. Then
Tr(Mn(z)) = (−2n+ 2)z−1 + 2zTr
(
T
(
n− 1; z2
))
= 0.
From the expression of T (n; z) (2.17) and the equation (2.24), we conclude entries of Mn(z) are
rational in z. ■
3 From the Lax Pair to the discrete Painlevé II hierarchy
In this section, we study the compatibility condition associated to the linear system (2.15). This
first allows us to reconstruct completely the matrix T (n; z) and then to obtain an explicit 2N
order discrete equation for xn which corresponds to equation (1.10).
3.1 The symmetry in the compatibility condition
We study the consequences of the symmetry (2.19) for the matrix T (n; z) on the compatibil-
ity condition for the Lax pair introduced in Proposition 2.11. More precisely, we show that,
thanks to the symmetry (2.19), the compatibility condition contains an overdetermined system
of equations.
We recall that the compatibility condition reads as
σ+ − T (n+ 1; z)U(n; z) + U(n; z)T (n; z) = 0, (3.1)
where we have to replace U(n; z) as in (2.16) and T (n; z) as
T (n; z) =
N+1∑
k=1
Tk(n)z
N−k +
2N+1∑
k=N+2
−K(n)T2N+2−k(n)K(n)−1zN−k, (3.2)
and with the coefficient TN+1(n) satisfying equation (2.21).
18 T. Chouteau and S. Tarricone
Lemma 3.1. The compatibility condition (3.1), for U(n; z), T (n; z) as described above, corre-
sponds to the following system
T1(n+ 1)σ+ − σ+T1(n) = 0,
Tj+1(n+ 1)σ+ − σ+Tj+1(n) + Tj(n+ 1)U0(n)− U0(n)Tj(n) = σ+δj,N , j = 1, . . . , N,
TN+1(n) = −K(n)TN+1(n)K(n)−1 + nI2.
Proof. The compatibility condition (3.1), after replacing U(n; z), T (n; z) of the prescribed
form, involves powers of z from N to −N − 1. Imposing that the coefficients of each of these
powers of z is identically zero, we obtain the following equations:
zN : T1(n+ 1)σ+ − σ+T1(n) = 0, (3.3)
zN−j , j = 1, . . . , N :
Tj+1(n+ 1)σ+ − σ+Tj+1(n) + Tj(n+ 1)U0(n)− U0(n)Tj(n) = σ+δj,N, (3.4)
z−1 : TN+1(n+ 1)U0(n)− U0(n)TN+1(n)−K(n+ 1)TN (n+ 1)K(n+ 1)−1σ+
+ σ+K(n)TN (n)K(n)−1 = 0, (3.5)
zN−j , j = N + 2, . . . , 2N :
−K(n+ 1)T2N+1−j(n+ 1)K(n+ 1)−1σ+ + σ+K(n)T2N+1−j(n)K(n)−1
+ U0(n)K(n)T2N+2−j(n)K(n)−1
−K(n+ 1)T2N+2−j(n+ 1)K(n+ 1)−1U0(n) = 0, (3.6)
z−N−1 : −K(n+ 1)T1(n+ 1)K(n+ 1)−1U0(n) + U0(n)K(n)T1(n)K(n)−1 = 0. (3.7)
With the change of indices 2N+1−j = k ⇐⇒ k = 2N+1−j = N−1, . . . , 1, the equation (3.6)
becomes:
−K(n+ 1)Tk(n+ 1)K(n+ 1)−1σ+ + σ+K(n)Tk(n)K(n)−1
−K(n+ 1)Tk+1(n+ 1)K(n+ 1)−1U0(n) + U0(n)K(n)Tk+1(n)K(n)−1 = 0. (3.8)
We now show that equations (3.5), (3.6), (3.7) are equivalent to the first ones (3.3), (3.4) thanks
to the symmetry of the coefficients Tk(n) given in (2.20) together with the equation for TN+1(n),
already obtained in (2.21).
To start with, we notice the following relations:
Ũ0(n) := K(n+ 1)−1U0(n)K(n) = σ+,
σ̃(n) := K(n+ 1)−1σ+K(n) = U0(n),
deduced by using multiple times relation (2.9), namely x2n +
κ2
n−1
κ2
n
= 1.
1. Let us consider first the equation (3.7) obtained from the coefficient of the term z−N−1.
Multiplying by K(n+ 1)−1 to the left and by K(n) to the right, we obtain
−T1(n+ 1)Ũ0(n) + Ũ0(n)T1(n) = 0,
that is exactly (3.3).
2. Let us consider now equations (3.8), obtained from the coefficients of the term zN−j ,
j = N + 2, . . . , 2N . By multiplying by K(n+ 1)−1 to the left and by K(n) to the right as
before, we obtain the equations for k = N − 1, . . . , 1
−Tk(n+ 1)σ̃(n) + σ̃(n)Tk(n)− Tk+1(n+ 1)Ũ0(n) + Ũ0(n)Tk+1(n) = 0,
which is exactly equation (3.4) for j = 1, . . . , N − 1.
Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy 19
3. The last equation is (3.5) obtained from the coefficient of the term z−1. We multiply,
again, by K(n+ 1)−1 to the left and by K(n) to the right, and we get
K(n+ 1)−1TN+1(n+ 1)K(n+ 1)Ũ0(n)− Ũ0(n)K(n)−1TN+1(n)K(n)
− TN (n+ 1)σ̃(n) + σ̃(n)TN (n) = 0,
and then we replace the symmetry for the term TN+1(n) namely the equation (2.21) (that
indeed it has not be used until now)
−TN+1(n+ 1)Ũ0(n) + Ũ0(n)TN+1(n) + Ũ0(n)− TN (n+ 1)σ̃(n) + σ̃(n)TN (n) = 0.
And this is again exactly equation (3.4), for j = N .
Thus the compatibility condition (3.1) is reduced to the equations in the statement, namely
equations (3.3), (3.4), (2.21). ■
Now, we use equations (3.3), (3.4) together with the initial condition for T1(n) given in (2.18),
to recursively find the coefficients Tk(n), for k = 1, . . . , N+1, in terms of the xn±j , j = 1, . . . , N .
With the coefficients Tk(n) computed in such a way, the symmetry for TN+1(n), i.e., equa-
tion (2.21), once TN+1(n) is determined, provides an actual discrete equation for xn of order 2N ,
that is what we call the higher order analogue of the discrete Painlevé II equation (that coincide
for N = 1, 2 to the ones already appeared in [1, 6, 11]).
3.2 The recursion
In this subsection, we explain how equations (3.3), (3.4) resulting from the compatibility con-
dition (3.1) can be used to find recursively (in k) all the coefficients Tk(n), k = 1, . . . , N + 1 of
T (n; z).
Lemma 3.2. For every i = 1, . . . , N , starting from the initial condition (2.18) T1(n) =
θN
2 σ3,
we have
Ti+1,12(n) = xn+1
(
2∆−1 + I
)(xn+1
vn+1
Ti,21(n+ 1)− xnTi,12(n)
)
+ vn+1Ti,12(n+ 1)
− xnxn+1Ti,12(n),
Ti+1,21(n+ 1) = xnvn+1
(
2∆−1 + I
)(xn+1
vn+1
Ti,21(n+ 1)− xnTi,12(n)
)
+ vn+1Ti,21(n)
− xnxn+1Ti,21(n+ 1),
Ti+1,11(n) = −Ti+1,22(n) + nδi,N = ∆−1
(
−xn+1
vn+1
Ti+1,21(n+ 1) + xnTi+1,12(n)
)
+ nδi,N ,
where
∆: Ti(n) → Ti(n+ 1)− Ti(n), (3.9)
vn := 1− x2n, (3.10)
Proof. We rewrite equations (3.3), (3.4) for i = 1, . . . , N , entry by entry. For the first one,
we have{
T1,11(n+ 1)− T1,11(n) = 0,
T1,12(n) = T1,21(n+ 1) = 0.
20 T. Chouteau and S. Tarricone
This is satisfied by T1(n) given in (2.18). For the second one, for any 1 ⩽ i ⩽ N we have the
four equations:
Ti+1,11(n+ 1)− Ti+1,11(n) = −Ti,11(n+ 1)xnxn+1 + Ti,12(n+ 1)
(
1− x2n+1
)
xn
+ xnxn+1Ti,11(n)− xn+1Ti,21(n) + δi,N ,
Ti+1,12(n) = −xn+1Ti,11(n+ 1) + Ti,12(n+ 1)
(
1− x2n+1
)
− xnxn+1Ti,12(n) + xn+1Ti,22(n),
Ti+1,21(n+ 1) = −Ti,21(n+ 1)xnxn+1 + Ti,22(n+ 1)xn
(
1− x2n+1
)
− Ti,11(n)xn
(
1− x2n+1
)
+
(
1− x2n+1
)
Ti,21(n),
0 = Ti,21(n+ 1)xn+1− Ti,22(n+ 1)
(
1− x2n+1
)
− xn
(
1− x2n+1
)
Ti,12(n) + Ti,22(n)
(
1− x2n+1
)
.
Using the notations introduced in (3.9), (3.10), the previous equations with 1 ⩽ i ⩽ N
become
∆Ti+1,11(n) = −xnxn+1∆Ti,11(n) + xnvn+1Ti,12(n+ 1)− xn+1Ti,21(n) + δi,N , (3.11)
Ti+1,12(n) = −xn+1Ti,11(n+ 1)+ vn+1Ti,12(n+ 1)− xnxn+1Ti,12(n)+ xn+1Ti,22(n), (3.12)
Ti+1,21(n+ 1) = −xnxn+1Ti,21(n+ 1) + xnvn+1Ti,22(n+ 1)− xnvn+1Ti,11(n)
+ vn+1Ti,21(n), (3.13)
vn+1∆Ti,22(n) = xn+1Ti,21(n+ 1)− xnvn+1Ti,12(n). (3.14)
From these equations, we see that in order to obtain the diagonal terms, there is a “discrete
integration” to perform, while the off-diagonal terms are directly determined from the previous
ones. Moreover, we can rewrite the four equation as only two equations involving only the off-
diagonal terms. Indeed, because of Tr(T (n; z)) = nz−1, Ti,11(n, z) = −Ti,22(n, z) for 1 ⩽ i ⩽ N .
Thus (3.14) can be written as
vn+1∆Ti,11(n) = −xn+1Ti,21(n+ 1) + xnvn+1Ti,12(n).
Formally, 1 ⩽ i ⩽ N ,
Ti,11(n) = −Ti,22(n) = ∆−1
(
−xn+1
vn+1
Ti,21(n+ 1) + xnTi,12(n)
)
, (3.15)
which still holds for i = N +1 up to adding the “constant” n on the right hand side. Using this
in (3.12) and (3.13), we obtain:
Ti+1,12(n) = xn+1
(
2∆−1 + I
)(xn+1
vn+1
Ti,21(n+ 1)− xnTi,12(n)
)
+ vn+1Ti,12(n+ 1)
− xnxn+1Ti,12(n),
Ti+1,21(n+ 1) = xnvn+1
(
2∆−1 + I
)(xn+1
vn+1
Ti,21(n+ 1)− xnTi,12(n)
)
+ vn+1Ti,21(n)
− xnxn+1Ti,21(n+ 1). ■
We notice that, defining the discrete recursion operator
L
(
un
yn
)
=
xn+1
(
2∆−1 + I
)(xn+1
vn+1
yn − xnun
)
+ (vn+1(∆ + I)− xnxn+1)un
xnvn+1
(
2∆−1+I
)(xn+1
vn+1
yn−xnun
)
+
(
vn+1(∆ + I)−1−xnxn+1
)
yn
, (3.16)
Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy 21
we can rewrite the two equations for the off-diagonal entries of Ti(n) obtained above as(
Ti+1,12(n)
Ti+1,21(n+ 1)
)
= L
(
Ti,12(n)
Ti,21(n+ 1)
)
, 1 ⩽ i ⩽ N. (3.17)
And, recursively we obtain(
TN+1,12(n)
TN+1,21(n+ 1)
)
= LN
(
0
0
)
. (3.18)
This procedure allows to construct the whole matrix T (n; z), starting from the initial condition
T1(n) =
θN
2 σ3 and iterating the operator L we obtain off diagonal terms of T (n; z) and compute
diagonal one with equation (3.15). Below we implemented this method to find the matrix T (n; z)
in the first few cases N = 1, 2.
Example 3.3. In the case N = 1, the matrix T (n; z) = T1(n)+T2(n)z
−1+T3(n)z
−2. Knowing
T1(n), we only have to find T2(n) using the recurrence relation given from the compatibility, i.e.,
equations (3.11), (3.12), (3.13) for i = 1. Since: T1,12(n) = T1,21(n) = 0, and T1,11(n) = θN/2 =
−T1,22(n), we have
T2,11(n) = n,
T2,12(n) = −xn+1(T1,11(n+ 1) + T1,11(n)) = −θ1xn+1,
T2,21(n+ 1) = xnvn+1(T1,22(n+ 1) + T1,22(n)) = −θ1xnvn+1,
and T2,22(n) = n−T2,11(n) = 0. Moreover, the symmetry which reflects terms of T (n; z) two by
two gives T3(n) = −K(n)T1(n)K(n). Thus the Lax matrix for N = 1 is
T (n; z) =
θ1
2
(
1 0
0 −1
)
+
1
z
(
n −θ1xn+1
−θ1vnxn−1 0
)
+
θ1
z2
(
1
2 − x2n xn
vnxn x2n − 1
2
)
.
Example 3.4. In the case N = 2, the matrix T (n; z) = T1(n)z+T2(n)+T3(n)z
−1+T4(n)z
−2+
T5(n)z
−3. This time we have to find T2(n) (that will be almost the same as before) and also T3(n)
using the recurrence relation given from the compatibility, i.e., equations (3.11), (3.12), (3.13)
for i = 1 and 2. First we find T2(n) (i = 1 above), we have
T2,11(n) =
θ1
2
,
T2,12(n) = −xn+1(T1,11(n+ 1) + T1,11(n)) = −θ2xn+1,
T2,21(n+ 1) = xnvn+1(T1,22(n+ 1) + T1,22(n)) = −θ2xnvn+1,
and T2,22(n) = −T2,11 = − θ1
2 .
Then we consider the equation for i = 2 and find T3(n). We have
∆T3,11(n) = xnvn+1(−θ2xn+2)− xn+1(−θ2xn−1vn) + 1 =⇒ T3,11(n) = n− θ2xn−1xn+1vn,
T3,12(n) = −θ1xn+1 − θ2
(
vn+1xn+2 − xnx
2
n+1
)
,
T3,21(n+ 1) =
(
−θ1xn − θ2
(
vnxn−1 − x2nxn+1
))
vn+1,
T3,22(n) = n− T3,11(n) = θ2xn−1xn+1vn.
Finally, we take T4(n) = −K(n)T2(n)K(n) and T5(n) = −K(n)T1(n)K(n). Thus the Lax
matrix for N = 2 is
T (n; z) = z
θ2
2
(
1 0
0 −1
)
+
(
θ1
2 −θ2xn+1
−θ2xn−1vn − θ1
2
)
22 T. Chouteau and S. Tarricone
+
1
z
(
n− θ2xn−1xn+1vn −θ1xn+1 − θ2
(
vn+1xn+2 − xnx
2
n+1
)(
−θ1xn−1 − θ2
(
vn−1xn−2 − xnx
2
n−1
))
vn θ2xn−1xn+1vn
)
+
1
z2
(
−θ2vn(xnxn−1 + xnxn+1) +
θ1
2
(
vn − x2n
)
−θ2
(
vnxn−1 + x2nxn+1
)
−θ2
(
vnxn+1 + x2nxn−1
)
vn θ2vn(xnxn−1 + xnxn+1)− θ1
2
(
vn − x2n
))
+
θ2
z3
(
1
2 − x2n xn
vnxn x2n − 1
2
)
.
Now that we have reconstructed the whole matrix T (n; z) in terms of xn±j , j = −N, . . . , N
we are left with the equation that TN+1(n) has to satisfy, namely (2.21). We now show that
actually this coincide with only one scalar equation in TN+1,12 and TN+1,21. Indeed, entry by
entry it reads as the following system of four equations. From the off-diagonal entries
vnTN+1,12(n) = xn(TN+1,11(n)− TN+1,22(n))− TN+1,21(n),
vnTN+1,21(n) = xnvn(TN+1,11(n)− TN+1,22(n))− v2nTN+1,12(n)
(3.19)
and from the diagonal entries
n−
(
1 + x2n
)
TN+1,11(n)− vnTN+1,22(n) + xnTN+1,21(n) + xnvnTN+1,12(n) = 0,
n−
(
1 + x2n
)
TN+1,22(n)− vnTN+1,11(n)− xnTN+1,21(n)− xnvnTN+1,12(n) = 0.
We notice first that the four above equations are all the same. The first and the second equations
are the same up to a multiplication by vn. Using the relation TN+1,11(n) + TN+1,22(n) = n, we
can rewrite the third and the forth equations and obtain the same equation up to a sign. Finally,
multiplying by xn the first equation and using the relation TN+1,11(n)+TN+1,22(n) = n we obtain
the third one. Thus from now on we will refer only to (3.19), as for the remaining equation.
Using equation (3.14) and Tr(T (n; z)) = nz−1, we express equation (3.19) in function of
TN+1,12(n) and TN+1,21(n). Consider equation (3.19), with the identity Tr(TN+1(n)) = n, it is
rewritten as
vnTN+1,12(n) = xn(n− 2TN+1,22(n))− TN+1,21(n).
Equation (3.14) holds also for i = N + 1. It means it is possible to replace TN+1,22(n) in the
previous equation and obtain
nxn − vnTN+1,12(n)− TN+1,21(n)
− 2xn∆
−1
(
−xnTN+1,12(n) +
xn+1
vn+1
(∆ + I)TN+1,21(n)
)
= 0. (3.20)
3.3 The relation between Ti,12(n) and Ti,21(n)
The previous equation (3.2) depends on TN+1,12(n) and TN+1,21(n). The aim of this part is to
establish a connection between Ti,12(n) and Ti,21(n) to rewrite equation (3.2) just in function
of TN+1,12(n).
To accomplish this, we study the compatibility condition of C(n; z) := T (n; z)2 and U(n; z).
C(n; z) is rational in z with a pole of order −2N − 2 at 0. We write C(n; z) as
C(n; z) =
4N+1∑
i=1
Ci(n)z
2N−1−i (3.21)
with
Ci(n) :=
i∑
j=1
Tj(n)Ti+1−j(n) (3.22)
where C1(n) =
θ2N
4 I2.
Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy 23
In what follows we will need the following lemma:
Lemma 3.5. Diagonal coefficients of Ci(n) defined as in (3.22) satisfy the following equation:
∀1 ⩽ i ⩽ N, Ci,11(n) = Ci,22(n),
CN+1,11(n) = nθN + CN+1,22(n).
Proof. We express Ci,11(n) in function of Ti,kj(n). With the equation (3.22)
Ci,11(n) =
i∑
j=1
Tj,11(n)Ti+1−j,11(n) + Tj,12(n)Ti+1−j,21(n).
Then, the sum index change j = i− k + 1 leads to
Ci,11(n) =
i∑
k=1
Ti−k+1,11(n)Tk,11(n) + Ti−k+1,12(n)Tk,21(n).
Finally, with the relation Tr(T (n; z)) = nz−1,
� if 1 ⩽ i ⩽ N ,
Ci,11(n) =
i∑
k=1
Ti−k+1,22(n)Tk,22(n) + Tk,21(n)Ti−k+1,12(n) = Ci,22(n).
� if i = N + 1,
CN+1,11(n) = −2nT1,22(n) +
N+1∑
k=1
TN−k+2,22(n)Tk,22(n) + Tk,21(n)TN−k+2,12(n)
= nθN + CN+1,22(n). ■
We deduce the compatibility condition for C and U from the one for T and U .
Lemma 3.6. C(n; z) (3.21) and U(n; z) (2.16) satisfy the following compatibility condition:
C(n+ 1; z)U(n; z)− U(n; z)C(n; z) = T (n+ 1; z)σ+ + σ+T (n; z). (3.23)
Proof. Multiplying on the left (resp. on the right) equation (3.1) by T (n+1; z) (resp. T (n; z))
and summing these two equations leads to the result. ■
The left (resp. right) hand side of the equation in the previous lemma is an expression in
powers of z from z2N−1 to z−2N−2 (resp. from zN−1 to z−N−1). This equation leads to recursive
equation for Ci(n). We consider only expression in powers of z from z2N−1 to zN−1.
According to (3.1) and (3.23), ∀1 ⩽ i ⩽ N , Ci(n) and Ti(n) satisfy the same recursive
equation (see equations (3.11)–(3.14)). For i = N + 1, the equation is a bit different. The term
with δi,N is now multiplied by θN .
From these equations we deduce the following result.
Proposition 3.7. Let Ci(n) be as in (3.22). Then ∀1 ⩽ i ⩽ N ,
Ci(n) = αiI2 and CN+1(n) = θNnσ+ + αN+1I2.
24 T. Chouteau and S. Tarricone
Proof. We prove Proposition 3.7 by induction. For i = 1, we already know C1(n) =
θ2N
4 .
Suppose Ci(n) = αiI2 for i ⩽ N − 1. Ci+1(n) satisfies the following equations:
∆Ci+1,11(n) = −xnxn+1∆Ci,11(n) + xnvn+1Ci,12(n+ 1)− xn+1Ci,21(n) + θNδi,N ,
Ci+1,12(n) = −xn+1Ci,11(n+ 1) + vn+1Ci,12(n+ 1)− xnxn+1Ci,12(n) + xn+1Ci,22(n),
Ci+1,21(n+ 1) = −xnxn+1Ci,21(n+ 1) + xnvn+1Ci,22(n+ 1)− xnvn+1Ci,11(n)
+ vn+1Ci,21(n).
Using induction hypothesis,
∆Ci+1,11(n) = −0 · xnxn+1 + 0 · xnvn+1 − 0 · xn+1 + θNδi,N = θNδi,N ,
Ci+1,12(n) = −xn+1αi + 0 · vn+1 − 0 · xnxn+1 + xn+1αi = 0,
Ci+1,21(n+ 1) = −0 · xnxn+1 + xnvn+1αi − xnvn+1αi + 0 · vn+1 = 0.
From the first equation, we conclude Ci+1,11(n) = αi+1 if i ⩽ N − 1 (resp. CN+1,11(n) =
θNn+αN+1 if i = N) and according to Lemma 3.5 Ci+1,22(n) = αi+1 (resp. CN+1,22(n) = αN+1)
which concludes the proof. ■
From equation (3.22) and Proposition 3.7, we obtain
θNTi,11(n) = αi −
i−1∑
j=2
Tj,11(n)Ti−j+1,11(n) + Tj,12(n)Ti−j+1,21(n), (3.24)
θNTN+1,11(n) = nθN + αN+1 −
N∑
j=2
Tj,11(n)TN−j+2,11(n) + Tj,12(n)TN−j+2,21(n). (3.25)
With all this discussion on C(n; z) it is now possible to prove the following proposition.
Proposition 3.8. The following holds: ∀1 ⩽ i ⩽ N + 1, Ti,11(n), Ti,12(n) and Ti,21(n) are
polynomials in xn+j’s. Moreover, the following symmetries hold:
∃(Qi,n((un+j)1−i⩽j⩽i−1), Pi,n((un+j)1−i⩽j⩽i−1))
polynomials in un+j’s such that,
Ti,11(n) = Qi,n((xn+j)1−i⩽j⩽i−1) = Qi,n((xn−j)1−i⩽j⩽i−1),
Ti,12(n) = Pi,n((xn+j)1−i⩽j⩽i−1),
Ti,21(n) = vnPi,n((xn−j)1−i⩽j⩽i−1).
Proof. We prove this proposition by strong induction. For i = 1, T1(n) = θN
2 σ3, then defin-
ing Q1,n(un) := θN
2 , P1,n(un) := 0; T1,11(n) = Q1,n(xn), T1,12(n) = P1,n(xn) and T1,21(n) =
vnP1,n(xn).
Now suppose the property true for all j ∈ [[1, i]] with i ⩽ N and let (Qj,n, Pj,n)j⩽i be
polynomials in xn+j ’s satisfying the property. According to (3.24) (and (3.25) for i = N) and
strong induction hypothesis, Ti+1(n) is a polynomial in xn+j ’s and the invariance when you
exchange xn+j by xn−j holds.
Because of equation (3.12) (resp. equation (3.13)) and of induction hypothesis, there exists
Pi+1,n((un+j)−i⩽j⩽i) (resp. P̃i+1,n((un+j)−i⩽j⩽i)) a polynomial such that
Ti+1,12(n) = Pi+1,n((xn+j)−i⩽j⩽i),
Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy 25
respectively,
Ti+1,21(n) = P̃i+1,n((xn+j)−i⩽j⩽i).
Now we establish the link between Pi+1,n and P̃i+1,n. According to equation (3.12) and the
relation Tr(T (n; z)) = nz−1,
Pi+1,n
(
(xn+j)
i
j=−i
)
= − xn+1Qi,n+1
(
(xn+j)
i−2
j=−i
)
+ vn+1Pi,n+1
(
(xn+j)
i−2
j=−i
)
− xnxn+1Pi,n
(
(xn+j)
i−1
j=1−i
)
− xn+1Qi,n
(
(xn+j)
i−1
j=1−i
)
.
Then
vnPi+1,n
(
(xn−j)
i
j=−i
)
= vn
(
− xn−1Qi,n−1
(
(xn−j)
i−2
j=−i
)
+ vn−1Pi,n−1
(
(xn−j)
i−2
j=−i
)
− xnxn−1Pi,n
(
(xn−j)
i−1
j=1−i
)
− xn−1Qi,n
(
(xn−j)
i−1
j=1−i
))
.
From induction hypothesis and Tr(T (n; z)) = nz−1
vnPi+1,n
(
(xn−j)
i
j=−i
)
= − xn−1vnTi,11(n− 1) + vnTi,21(n− 1) + xn−1xnTi,21(n)
+ xn−1vnTi,22(n).
According to equation (3.13),
vnPi+1,n
(
(xn−j)
i
j=−i
)
= Ti+1,21(n+ 1).
Then
vnPi+1,n
(
(xn−j)
i
j=−i
)
= P̃i+1,n
(
(xn+j)−i⩽j⩽i
)
and this concludes the proof. ■
Define C[(xj)j∈[[0,2n]]] and the transformation
Permn : C[(xj)j∈[[0,2n]]] −→ C[(xj)j∈[[0,2n]]],
P ((xn+j)−n⩽j⩽n) 7−→ P ((xn−j)−n⩽j⩽n).
From the previous proposition,
Ti,21(n) = vnPermn(Ti,12(n)). (3.26)
Remark 3.9. As a consequence of the Proposition 3.8, the equation (3.19) is a polynomial
in xn+j ’s and is invariant when you apply Permn to this equation because Perm2
n = Id and
Permnvn = vnPermn.
We use the link we established in Proposition 3.8 between Ti,12(n) and Ti,21(n) to rewrite the
operator L (3.16) as a scalar operator:
L(un) :=
(
xn+1
(
2∆−1 + I
)
((∆ + I)xnPermn − xn) + vn+1(∆ + I)− xnxn+1
)
un. (3.27)
Finally, collecting all the results from the previous sections, we state and proof the following
theorem.
Theorem 3.10. The system (2.15), with T (n; z) of the form (3.2) and coefficient TN+1(n) satis-
fying the symmetry condition (2.21), is a Lax pair for the N -th higher order discrete Painlevé II
equation and the equation is given by the expression:
nxn +
(
2xn∆
−1(xn − (∆ + I)xnPermn)− vn − vnPermn
)
TN+1,12(n) = 0, (3.28)
where TN+1,12(n) = LN (0) with L as in (3.27).
26 T. Chouteau and S. Tarricone
Proof. Replacing TN+1,21(n) with the relation (3.26), equation (3.2) now reads as
nxn +
(
2xn∆
−1(xn − (∆ + I)xnPermn)− vn − vnPermn
)
TN+1,12(n) = 0.
Equations (3.17) and (3.18) with the relation (3.26) reduce to
Ti+1,12(n) = L(Ti,12(n)) and TN+1,12(n) = LN (0),
which concludes the proof. ■
The next two examples explain for N = 1, 2 how to compute explicitly equation (3.28).
Example 3.11. Using the expression defined in Theorem 3.10, we compute the first equa-
tion (1.13) and the second (1.14).
For N = 1: First we compute T2,12(n) with the operator L (3.27):
T2,12(n) = 2xn+1∆
−1(0) = −θ1xn+1,
where −θ1/2 is the integration constant.
Replacing T2,12(n) in equation (3.28),
nxn + vnθ1(xn+1 + xn−1) + 2xn∆
−1(θ1xnxn+1 − θ1xnxn+1) = 0.
Then
(n+ α)xn + θ1vn(xn+1 + xn−1) = 0.
This equation is the same as equation (1.13) if we choose the integration constant α to be zero.
For N = 2: We compute T3,12(n). Computations are the same for T2,12(n) except for the
integration constant, T2,12(n) = −θ2xn+1.
T3,12(n) = L(T2,12(n)) =
(
xnx
2
n+1 − vn+1xn+2
)
θ2
+ xn+1
(
2∆−1 + I
)
(−θ2xnxn+1 + θ2xnxn+1)
Then T3,12(n) = θ2
(
xnx
2
n+1 − vn+1xn+2
)
− θ1xn+1.
Replacing T3,12(n) in equation (3.28),
(n+ α)xn + θ2vn
(
vn+1xn+2 + vn−1xn−2 − xn(xn+1 + xn−1)
2
)
+ θ1vn(xn+1 + xn−1) = 0
which is the same equation as (1.14).
We finally conclude the work by noticing that Theorem 3.10 together with Corollary 2.8 give
the proof of Theorem 1.2.
Remark 3.12. In our setting, the fixed N ≥ 1 define the order (2N) of the discrete equation
solved by xn, the quantity related to the Toeplitz determinants Dn. An alternative approach
could be to leave N variate and consider it as a second discrete variable for xn. In effect, this is
done in [19], where the authors consider orthogonal polynomials on the real line, w.r.t. a weight
ρ(λ;N)dλ and where the dependence on an integer parameter N is such that ρ(λ;N + 1) =
λρ(λ;N). In this case the relevant quantities to consider (related to the Hankel determinants)
are the coefficients of the three terms recurrence relation satisfied by these polynomials. The
authors there proved that these quantities solve (up to some change of variables) the discrete-
time Toda molecule equation, a coupled system of discrete equations in the two variables n, N .
The result deeply relies on the quasi-periodic condition satisfied by the weight ρ. Back to our
setting, the measure we have for our orthogonal polynomials on the unit circle is such that
dµ(λ;N + 1) = e
∑N+1
j=1
θj
j
(eiλj+e−iλj)dλ
2π
= e
θN+1
N+1
(eiλ(N+1)+e−iλ(N+1)) dµ(λ;N).
This relation does not seem as promising as the one for ρ for the study of the N -dependence,
but it is another point that we could further investigate.
Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy 27
A The continuous limit
This appendix contains further computations for the continuous limit of the equations of the
discrete Painlevé II hierarchy (1.10) in the first cases N = 1, 2, 3. To obtain it, we follow the
scaling limit given in [5, Theorem 1] as already recalled in the introduction.
The case N = 1. Notice that in this case we recover the same computation done in [6,
Chapter 9]. We consider equation (1.13) written as
xn+1 + xn−1 +
nxn
θ1
(
1− x2n
) = 0
in which the only parameter appearing is θ1 = θ. Following the scaling limit of [5, Theorem 1],
in the case N = 1, we have
b = 2, d = 1 and xn = (−1)nθ−
1
3u(t) with t = (n− 2θ)θ−
1
3 .
Now, for θ → +∞, we compute
xn±1 ∼ (−1)n+1θ−
1
3u
(
t± θ−
1
3
)
∼ (−1)n+1θ−
1
3
(
u(t)± θ−
1
3u′(t) +
θ−
2
3
2
u′′(t) +O
(
θ−1
))
,
that gives
xn+1 + xn−1 ∼ (−1)n+12θ−
1
3u(t) + (−1)n+1θ−1u′′(t) +O
(
θ−1
)
.
The other term appearing in the discrete Painlevé II equation gives instead
nxn
θ1
(
1− x2n
) ∼
(
2θ + tθ
1
3
)
(−1)nθ−
1
3u(t)θ−1
(
1 + θ−
2
3u2(t) +O
(
θ−1
))
∼ (−1)n2θ−
1
3u(t) + (−1)nθ−1
(
tu(t) + 2u3(t)
)
+O
(
θ−1
)
.
Thus equation (1.8) in this scaling limit gives at the first order (coefficient of θ−1) the second
order differential equation
u′′(t)− tu(t)− 2u3(t) = 0,
which coincides indeed with the Painlevé II equation.
The case N = 2. We consider equation (1.14), with the parameters θ1, θ2 rescaled as
θ1 = θ, θ2 =
θ
4 . It reads as
nxn(
1− x2n
) + θ(xn+1 + xn−1)
+
θ
4
(
xn+2
(
1− x2n+1
)
+ xn−2
(
1− x2n−1
)
− xn(xn+1 + xn−1)
2
)
= 0 (A.1)
and this time we consider the following scaling limit (case N = 2 in [5, Theorem 1])
b =
3
2
, d = 4 and xn = (−1)nθ−
1
5 4
1
5u(t) with t =
(
n− 3
2
θ
)
θ−
1
5 4
1
5 .
For θ → +∞, similar computations gives the fourth order differential equation
tu(t) + 6u(t)5 − 10u(t)u′(t)2 − 10u(t)2u′′(t) + u′′′′(t) = 0
28 T. Chouteau and S. Tarricone
which corresponds to the second equation of the Painlevé II hierarchy. Detailed computations to
obtain certain terms from the previous equation are given below. We begin with the expansion
of the first term in equation (A.1):
nxn(
1− x2n
) ∼
(
3
2
θ + 4−
1
5 θ
1
5 t
)
(−1)nθ−
1
5 4
1
5u(t)
(
1 + 4
2
5 θ−
2
5u2(t) + 4
4
5 θ−
4
5u4(t) +O
(
θ−1
))
∼ (−1)n
(
3
2
4
1
5 θ
4
5u(t) +
3
2
4
3
5 θ
2
5u(t)3 + tu(t) + 6u(t)5 +O
(
θ−
1
5
))
.
Computing expansions of xn±1, xn±2 as θ → ∞, we obtain
xn±1 ∼ (−1)n+14
1
5 θ−
1
5u
(
t± 4
1
5 θ−
1
5
)
∼ (−1)n+14
1
5 θ−
1
5
×
(
u(t)± 4
1
5 θ−
1
5u′(t) +
4
2
5 θ−
2
5
2
u′′(t)± 4
3
5 θ−
3
5
6
u′′′(t) +
4
4
5 θ−
4
5
24
u′′′′(t) +O
(
θ−1
))
,
xn±2 ∼ (−1)n4
1
5 θ−
1
5u
(
t± 2θ−
1
5 4
1
5
)
∼ (−1)n4
1
5 θ−
1
5
×
(
u(t)± 4
1
5 2θ−
1
5u′(t) + 4
7
5 θ−
2
5u′′(t)± 4
8
5 2θ−
3
5
3
u′′′(t) +
4
9
5 θ−
4
5
3
u′′′′(t) +O
(
θ−1
))
that gives for the second term of equation (A.1)
θ(xn+1 + xn−1) ∼ (−1)n+1
(
4
1
5 2θ
4
5u(t) + 4
3
5 θ
2
5u′′(t) +
1
3
u′′′′(t) +O
(
θ−
1
5
))
.
Some linear and nonlinear terms appear with the expansion of the third term of equation (A.1).
The linear one is
θ
4
(xn+2 + xn−2) ∼ (−1)n
(
4
1
5 θ
4
5
1
2
u(t) + 4
3
5 θ
2
5u′′(t) +
4
3
u′′′′(t) +O
(
θ−
1
5
))
.
Nonlinear ones are
θ
4
xn(xn+1 + xn−1)
2 ∼ (−1)nu(t)
(
4
3
5 θ
2
5u(t)2 + 4u(t)u′′(t) +O
(
θ−
1
5
))
,
θ
4
xn±2x
2
n±1 ∼ (−1)n
(
4−
2
5 θ
2
5u(t)3 ± 4
4
5 θ
1
5u(t)2u′(t) + 3u(t)2u′′(t) + 5u(t)u′(t)2
)
.
From these computations, we see that we recover exactly
tu(t) + 6u(t)5 − 10u(t)u′(t)2 − 10u(t)2u′′(t) + u′′′′(t) = 0.
The case N = 3. We consider equation (1.15) with the parameters θ1, θ2, θ3 rescaled as
θ1 = θ, θ2 =
2θ
5 , θ3 =
θ
15 and rewritten as
nxn
θ
(
1− x2n
) + (xn+1 + xn−1) +
2
5
(
xn+2
(
1− x2n+1
)
+ xn−2
(
1− x2n−1
)
− xn(xn+1 + xn−1)
2
)
+
1
15
(
x2n(xn+1 + xn−1)
3 + xn+3
(
1− x2n+2
)(
1− x2n+1
)
+ xn−3
(
1− x2n−2
)(
1− x2n−1
))
+
1
15
(
−2xn(xn+1 + xn−1)
(
xn+2
(
1− x2n+1
)
+ xn−2
(
1− x2n−1
))
− xn−1x
2
n−2
(
1− x2n−1
))
+
1
15
(
−xn+1x
2
n+2
(
1− x2n+1
)
− xn+1xn−1(xn+1 + xn−1)
)
= 0.
Finally, we consider the following scaling limit (case N = 3 of [5, Theorem 1])
b =
4
3
, d = 15 and xn = (−1)nθ−
1
7 15
1
7u(t) with t =
(
n− 4
3
θ
)
θ−
1
7 15
1
7 .
Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy 29
Again, for θ → +∞ the asymptotic expansion of the equation above results at the first order
(coefficient of θ−1) into the sixth order differential equation
tu(t) + 20u(t)7 − 140u(t)3u′(t)2 − 70u(t)4u′′(t) + 70u′(t)2u′′(t) + 42u(t)u′′(t)2
+ 56u(t)u′(t)u′′′(t) + 14u(t)4u′′(t)− u′′′′′′(t) = 0,
which corresponds to the third equation in the Painlevé II hierarchy.
Remark A.1. Computations for N = 2 and N = 3 were performed with Maple/Mathematica.
Files are available on demand.
Acknowledgments
We acknowledge the support of the H2020-MSCA-RISE-2017 PROJECT No. 778010 IPaDE-
GAN and the International Research Project PIICQ, funded by CNRS. During the period
from November 2021 to October 2022, S.T. was supported also by the Fonds de la Recherche
Scientifique-FNRS under EOS project O013018F and based at the Institut de Recherche en
Mathématique et Physique of UCLouvain. The authors are grateful to Mattia Cafasso for the
inspiration given to work on this project and his guidance. The authors also want to thank the
referees of this paper for useful comments and suggestions. S.T. is also grateful to Giulio Ruzza
for meaningful conversations.
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1 Introduction
2 OPUC: the Riemann–Hilbert approach and a discrete Painlevé II Lax pair
2.1 Toeplitz determinants related to OPUC
2.2 Riemann–Hilbert problem associated to OPUC
2.3 A linear differential system for Psi(z)
2.4 Relation with the Cresswell–Joshi Lax pair
3 From the Lax Pair to the discrete Painlevé II hierarchy
3.1 The symmetry in the compatibility condition
3.2 The recursion
3.3 The relation between T_{i,12}(n) and T_{i,21}(n)
A The continuous limit
References
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| id | nasplib_isofts_kiev_ua-123456789-211913 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T00:46:36Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Chouteau, Thomas Tarricone, Sofia 2026-01-16T11:18:15Z 2023 Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy. Thomas Chouteau and Sofia Tarricone. SIGMA 19 (2023), 030, 30 pages 1815-0659 2020 Mathematics Subject Classification: 33E17; 33C47; 35Q15 arXiv:2211.16898 https://nasplib.isofts.kiev.ua/handle/123456789/211913 https://doi.org/10.3842/SIGMA.2023.030 Solutions of the discrete Painlevé II hierarchy are shown to be in relation to a family of Toeplitz determinants describing certain quantities in multicritical random partition models, for which the limiting behavior has been recently considered in the literature. Our proof is based on the Riemann-Hilbert approach for the orthogonal polynomials on the unit circle related to the Toeplitz determinants of interest. This technique allows us to construct a new Lax pair for the discrete Painlevé II hierarchy that is then mapped to the one introduced by Cresswell and Joshi. We acknowledge the support of the H2020-MSCA-RISE-2017 PROJECT No. 778010 IPaDEGAN and the International Research Project PIICQ, funded by CNRS. During the period from November 2021 to October 2022, S.T. was also supported by the Fonds de la Recherche Scientifique-FNRS under the EOS project O013018F and based at the Institut de Recherche en Mathématique et Physique of UCLouvain. The authors are grateful to Mattia Cafasso for the inspiration given to work on this project and his guidance. The authors also want to thank the referees of this paper for their useful comments and suggestions. S.T. is also grateful to Giulio Ruzza for meaningful conversations. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy Article published earlier |
| spellingShingle | Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy Chouteau, Thomas Tarricone, Sofia |
| title | Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy |
| title_full | Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy |
| title_fullStr | Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy |
| title_full_unstemmed | Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy |
| title_short | Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy |
| title_sort | recursion relation for toeplitz determinants and the discrete painlevé ii hierarchy |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211913 |
| work_keys_str_mv | AT chouteauthomas recursionrelationfortoeplitzdeterminantsandthediscretepainleveiihierarchy AT tarriconesofia recursionrelationfortoeplitzdeterminantsandthediscretepainleveiihierarchy |