Exceptional Legendre Polynomials and Confluent Darboux Transformations
Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of ''...
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| description | Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of ''exceptional'' degrees. In this paper, we introduce a new construction of multi-parameter exceptional Legendre polynomials by considering the isospectral deformation of the classical Legendre operator. Using confluent Darboux transformations and a technique from inverse scattering theory, we obtain a fully explicit description of the operators and polynomials in question. The main novelty of the paper is the novel construction that allows for exceptional polynomial families with an arbitrary number of real parameters.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 016, 19 pages
Exceptional Legendre Polynomials
and Confluent Darboux Transformations
Maŕıa Ángeles GARCÍA-FERRERO †1, David GÓMEZ-ULLATE †2†3 and Robert MILSON †4
†1 Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg,
Im Neunheimer Feld 205, 69120 Heidelberg, Germany
E-mail: garciaferrero@uni-heidelberg.de
†2 Departamento de Ingenieŕıa Informática, Escuela Superior de Ingenieŕıa,
Universidad de Cádiz, 11519 Puerto Real, Spain
E-mail: david.gomezullate@uca.es
†3 Departamento de F́ısica Teórica, Universidad Complutense de Madrid, 28040 Madrid, Spain
†4 Department of Mathematics and Statistics, Dalhousie University,
Halifax, NS, B3H 3J5, Canada
E-mail: rmilson@dal.ca
Received September 22, 2020, in final form February 03, 2021; Published online February 20, 2021
https://doi.org/10.3842/SIGMA.2021.016
Abstract. Exceptional orthogonal polynomials are families of orthogonal polynomials that
arise as solutions of Sturm–Liouville eigenvalue problems. They generalize the classical fam-
ilies of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that
miss a finite number of “exceptional” degrees. In this paper we introduce a new construction
of multi-parameter exceptional Legendre polynomials by considering the isospectral defor-
mation of the classical Legendre operator. Using confluent Darboux transformations and
a technique from inverse scattering theory, we obtain a fully explicit description of the oper-
ators and polynomials in question. The main novelty of the paper is the novel construction
that allows for exceptional polynomial families with an arbitrary number of real parameters.
Key words: exceptional orthogonal polynomials; Darboux transformations; isospectral de-
formations
2020 Mathematics Subject Classification: 33C47; 34L10; 34A05
1 Introduction and main results
Exceptional orthogonal polynomials (XOPs) are complete families of orthogonal polynomials
that arise as eigenfunctions of a Sturm–Liouville eigenvalue problem [11]. XOPs are more
general than classical OPs, because the degree sequence of the polynomial family can have a finite
number of missing, “exceptional” degrees. As in the classical theory, XOPs fall into three broad
classes: Hermite, Laguerre and Jacobi, depending on whether the domain of orthogonality is
the full line, the half-line or a finite interval [7]. Unlike the classical case, the corresponding
exceptional second-order operator has rational rather than polynomial coefficients.
Exceptional polynomials appear in mathematical physics as bound states of exactly solvable
rational extensions [10, 24, 26] and exact solutions to Dirac’s equation [27]. They appear also in
connection with super-integrable systems [21, 25] and finite-gap potentials [17]. From a mathe-
matical point of view, the main results are concerned with the full classification of exceptional
polynomials [7, 12], properties of their zeros [14, 18, 20], and recurrence relations [5, 13, 22, 23].
At the time of this writing, the most general construction of exceptional Jacobi polyno-
mials [3, 6] involves a finite number of discrete parameters and is given in terms of a Wronskian-
mailto:garciaferrero@uni-heidelberg.de
david.gomezullate@uca.es
mailto:rmilson@dal.ca
https://doi.org/10.3842/SIGMA.2021.016
2 M.Á. Garćıa-Ferrero, D. Gómez-Ullate and R. Milson
like determinant of classical Jacobi polynomials, indexed by two partitions. The purpose of
this note is to show that the class of exceptional orthogonal polynomials is much richer than
previously thought. We do this by studying the class of exceptional Legendre polynomials, which
cannot be obtained using the standard approach of multi-step Darboux transformations indexed
by partitions. The main novelty of the new families is that they contain an arbitrary number
of continuous deformation parameters. Another innovation is the use of integral rather than
differential operators in the construction of the exceptional polynomials.
Definition 1.1. Let τ = τ(z) be a polynomial. We say that the operator
T̂ (τ) =
(
1− z2
) (
D2
z − 2
τz
τ
Dz +
τzz
τ
)
− 2zDz (1.1)
is an exceptional Legendre operator if there exist polynomials
{
P̂i(z)
}
i∈N0
and constants {λi}i∈N0 ,
where N0 = {0, 1, . . . }, such that
T̂ (τ)P̂i = λiP̂i
and such that the degree sequence
{
deg P̂i
}
i∈N0
is missing finitely many “exceptional” de-
grees [11].
Note that, in making this definition, we are not assuming that deg P̂i = i.
Remark 1.2. As a direct consequence of this definition, if τ(z) has no zeros on [−1, 1] and if
the eigenvalues are distinct, then the resulting eigenpolynomials are orthogonal relative to the
inner product∫ 1
−1
P̂i1(z)P̂i2(z)
τ(z)2
dz = 0, i1 6= i2. (1.2)
In this case, the eigenpolynomials
{
P̂i(z)
}
i∈N0
may define a complete orthogonal polynomial
system, which motivates the following definition.
Definition 1.3. Let τ(z) be a polynomial that does not vanish in [−1, 1]. The set
{
P̂i(z)
}
i∈N0
is a family of exceptional Legendre polynomials if
(i)
{
P̂i(z)
}
i∈N0
are eigenfunctions of a Sturm–Liouville problem in [−1, 1].
(ii)
{
deg P̂i
}
i∈N0
contains all but finitely many positive integers.
(iii) The polynomials
{
P̂i(z)
}
i∈N0
satisfy the orthogonality relation (1.2).
(iv) The polynomials
{
P̂i(z)
}
i∈N0
form a complete set in the Hilbert space L2
(
[−1, 1], τ−2dz
)
.
In other words, exceptional Legendre polynomials are just exceptional polynomials defined
in [−1, 1] with orthogonality weight W (z) = τ(z)−2, where τ(z) is a polynomial not vanishing
in [−1, 1]. It should be noted, however, that the standard construction of exceptional Jacobi
polynomials based on a multi-index determinant labelled by two partitions [6, equation (5.1)]
does not allow parameters α = β = 0 (see [3, equation (2.36)]). Thus, the construction of
exceptional Legendre polynomials requires a different approach, which we present in this paper.
It is known [7, Theorem 1.2] that every exceptional operator can be related to a classical
Bochner operator by a finite number of Darboux transformations. This is true in particular for
an exceptional operator having the form (1.1), where the degree of τ(z) is equal to the number
of exceptional degrees. The new exceptional polynomial families introduced in this paper do not
invalidate the classification result [7, Theorem 1.2], but rather they highlight the fact that the
Exceptional Legendre Polynomials and Confluent Darboux Transformations 3
full class of Darboux transformations leading to exceptional polynomials is larger than previously
thought.
As a matter of fact, we will consider in this paper a new class of exceptional operators that
are obtained from the classical Legendre operator
T := T̂ (1) =
(
1− z2
)
D2
z − 2zDz (1.3)
by the application of a finite number of confluent Darboux transformations (CDTs) [16], also
known as the “double commutator” method [9]. A CDT applied within a spectral gap of
a second-order self-adjoint operator allows to add one eigenvalue to the spectrum. We will
relate T to T̂ (τ) by a chain of CDTs, but the commutation procedure we consider is performed at
an existing eigenvalue. The resulting spectral transformation for every confluent pair “deletes”
an existing eigenvalue and then “adds” it back.1 The overall effect is that of an isospectral
transformation [19].
An important feature of confluent Darboux transformations is that every confluent pair of
transformations naturally introduces an extra deformation parameter. Known instances of ex-
ceptional Jacobi polynomials are indexed by discrete parameters and cannot be continuously
deformed into their classical counterparts. By contrast, after performing n CDTs on the classi-
cal Legendre operator (1.3) at distinct energy levels indexed by m = (m1, . . . ,mn) ∈ Nn0 we will
arrive at an exceptional Legendre operator
Tm(tm) = T̂ (τm(z; tm))
that depends on n real parameters tm = (tm1 , . . . , tmn) ∈ Rn. The polynomial eigenfunctions
of Tm(tm) are exceptional Legendre polynomials {Pm,i(z; tm)}i∈N0 , which depend on n real
parameters tm = (tm1 , . . . , tmn), and can be continuously deformed to the classical Legendre
polynomials by letting tm → 0.
Adapting certain methodologies from the theory of inverse scattering [1, 4, 29], we are able
to exhibit a determinantal representation of τm(z; tm) that is formally similar to the construc-
tion of KdV multi-solitons. The difference here is that, instead of dressing the zero potential,
we isospectrally deform a particular instance of the Darboux–Poschl–Teller potential [15] by
modifying the normalizations of a finite number of the corresponding bound states. Another
feature of our approach is that, rather than working with a Schrödinger operator, we remain in
a polynomial setting by utilizing the gauge and coordinate of the Legendre operator. The result
is a constructive procedure that can be easily implemented using a computer algebra system.
1.1 Notation and definitions
The base case of the construction is the classical Legendre operator T , shown in (1.3), and the
classical Legendre polynomials [30]
Pi(z) :=
2−i
i!
Di
z
(
z2 − 1
)i
= 2−i
i∑
k=0
(
i
k
)2
(z − 1)i−k(z + 1)k, i ∈ N0. (1.4)
These classical orthogonal polynomials do have degPi = i, they satisfy the eigenvalue relation
TPi = −i(i+ 1)Pi, i ∈ N0,
and they form an L2-complete orthogonal family relative to the inner product∫ 1
−1
Pi1(z)Pi2(z)dz =
2
2i1 + 1
δi1i2 , i1, i2 ∈ N0.
1This is true in a formal sense only, as the intermediate potential is singular.
4 M.Á. Garćıa-Ferrero, D. Gómez-Ullate and R. Milson
Before we can state the main results of the paper, we would like to fix some notation conven-
tions to be used throughout the paper. Bold symbols such as m, t or Q will represent tuples
of integers, real numbers or polynomials (one dimensional objects), while calligraphic symbols
like R will denote matrices. To access the components of a vector or tensor we will employ
square brackets, i.e., [R]k` denotes the (k, `) entry of R. In addition, given an n× n matrix R
and integers 1 ≤ k < ` ≤ n, we denote byMk,`(R) the (`−k+ 1)× (`−k+ 1) square submatrix
of R that includes the intersection of rows and columns from k to `.
If m = (m1, . . . ,mn) is an n-tuple and k ∈ {1, . . . , n}, we will denote by m〈k〉 the (n−1)-tuple
where the element [m]k is removed, i.e., m〈k〉 = (m1, . . . ,mk−1,mk+1, . . . ,mn), and by mk the
k-tuple formed by the first k elements of m, i.e., mk = (m1, . . . ,mk). In particular, we may
write explicitly mn instead of m whenever the context requires to emphasize the length of the
tuple, mostly in the proofs by induction or recurrence relations.
Associated to an n-tuple of integers m = (m1, . . . ,mn) ∈ Nn0 , we will define the n-tuple of real
parameters tm = (tm1 , . . . , tmn) ∈ Rn. Semicolons will be used to separate objects of different na-
ture. Commas will be used for tuple concatenation, e.g., if i1, . . . , ik ∈ N0, (m, i1, . . . , ik) denotes
the (n+k)-tuple (m1, . . . ,mn, i1, . . . , ik, ). Similarly, we have (tm, ti1 , . . . , tik) = t(m,i1,...,ik). Of-
ten, we will omit the parentheses when denoting 1-tuples, e.g., m1 instead of (m1). Finally, we
will use, depending on the context, the following notation for derivatives of a function f with
respect to z: Dzf , f ′ and fz.
With this notation in mind, we proceed to define the main objects of this paper.
Definition 1.4. Given an n-tuple m ∈ Nn0 and the associated tm ∈ Rn, we define Rm(z; tm)
as the n× n matrix with polynomial entries given by
[Rm(z; tm)]k` = δk` + tm`
Rmkm`
(z), k, ` ∈ {1, . . . , n},
where
Rmkm`
(z) :=
∫ z
−1
Pmk
(u)Pm`
(u)du, (1.5)
and Pi(z) denote the classical Legendre polynomials (1.4). We denote its determinant by
τm(z; tm) := detRm(z; tm). (1.6)
We define the n-tuple of polynomials
QT
m(z; tm) := τm(z; tm)Rm(z; tm)−1
(
Pm1(z), . . . , Pmn(z)
)T
, (1.7)
Finally, for i ∈ N0, we define the polynomials
Pm;i(z; tm) :=
[
Q(m,i)(z; t(m,i))
]
n+1
. (1.8)
Note that, by construction, τm(z; tm) is symmetric in m and Qm(z; tm) is equivariant with
respect to permutations of m. In addition, Pm;i(z; tm) is symmetric in m and does not depend
on ti since τ(m,i)(z; t(m,i))
[
R(m,i)(z; t(m,i))
−1]
n+1,j
correspond to the minors of the last column
of R(m,i)(z; t(m;i)), the only column where ti appears.
Exceptional Legendre Polynomials and Confluent Darboux Transformations 5
For example, for m1,m2 ∈ N0 we have
τm1(z; tm1) = 1 + tm1Rm1m1(z),
R(m1,m2)(z; t(m1,m2)) =
(
1 + tm1Rm1m1(z) tm2Rm1m2(z)
tm1Rm2m1(z) 1 + tm2Rm2m2(z)
)
,
τ(m1,m2)(z; t(m1,m2)) = 1 + tm1Rm1m1(z) + tm2Rm2m2(z)
+ tm1tm2
(
Rm1m1(z)Rm2m2(z)−R2
m1m2
(z)
)
,
Q(m1,m2)(z; t(m1,m2)) =
((
1 + tm2Rm2m2(z)
)
Pm1(z)− tm2Rm1m2(z)Pm2(z)(
1 + tm1Rm1m1(z)
)
Pm2(z)− tm1Rm2m1(z)Pm1(z)
)
,
Pm1;i(z; tm1) =
(
1 + tm1Rm1m1(z)
)
Pi(z)− tm1Rim1(z)Pm1(z), i ∈ N0.
After defining these objects, we are now ready to state the results.
1.2 Main results
The main result of this paper states that the polynomials {Pm;i(z; tm)}i∈N0 defined by (1.5)–(1.8)
are exceptional Legendre polynomials, provided the real parameters tm satisfy certain con-
straints to ensure that τm(z; tm) has constant sign on z ∈ [−1, 1].
Theorem 1.5. For m ∈ Nn0 , consider the operator
Tm(tm) := T̂ (τm(z; tm)),
given by (1.1) and (1.6). Then Tm(tm) is an exceptional Legendre operator that satisfies
Tm(tm)Pm;i(z; tm) = −i(i+ 1)Pm;i(z; tm), i ∈ N0, (1.9)
with Pm;i(z; tm) as in (1.8).
In light of (1.9), we may refer to Pm;i(z; tm), where i ∈ N0 varies and m and tm are fixed,
as exceptional Legendre polynomials. This requires according to Remark 1.2 and Definition 1.3
that τm(z; tm) does not vanish on [−1, 1]. The following theorem gives necessary and sufficient
conditions for this to be true. In that case, like their classical counterparts, the polynomials
{Pm;i(z; tm)}i∈N0 are orthogonal and complete.
Theorem 1.6. For m ∈ Nn0 with m1, . . . ,mn distinct, the polynomial τm(z; tm) in (1.6) has
no zeros on [−1, 1] if and only if
tmj > −mj −
1
2
, j ∈ {1, . . . , n}. (1.10)
If the above conditions hold, then {Pm;i(z; tm)}i∈N0 are exceptional Legendre polynomials with∫ 1
−1
Pm;i1(u; tm)Pm;i2(u; tm)
τm(z; tm)2
du =
2
1 + 2i1 + 2(δi1m1 + · · ·+ δi1mn)ti1
δi1i2 , i1, i2 ∈ N0.
and L2-completeness in [−1, 1] relative to the measure τm(z; tm)−2dz.
Remark 1.7. Note that we could reformulate the above result without the assumption that
m1, . . . ,mn are distinct. However, there is no extra benefit in doing this, as demonstrated by
Proposition 3.6 below. Assuming that the indices m1, . . . ,mn are all distinct does not entail any
loss of generality.
6 M.Á. Garćıa-Ferrero, D. Gómez-Ullate and R. Milson
The rest of the paper is organized as follows: in Section 2 we study exceptional Legendre
operators connected by a single step rational confluent Darboux transformation, which involves
in fact two Darboux transformations at the same factorization energy. We will iterate these
results in Section 3 to consider any number of confluent Darboux transformations and we will
relate this construction with the objects in Definition 1.4, thus yielding the proofs of the main
theorems. Finally, in Section 4 we give some explicit examples of the new exceptional Legendre
families.
2 One step confluent Darboux transformations
In this section we collect a number of relevant Propositions for the proofs of the Theorems 1.5
and 1.6. We introduce the concept of a rational confluent Darboux transformation and we show
that this transformation preserves the class of exceptional Legendre operators.
Before introducing rational confluent Darboux transformation, we recall 2-step ordinary
Darboux transformations between two operators T1 and T2 with rational coefficients. If A1,
A2, B1, B2 are first-order differential operators with rational coefficients, and λ1, λ2 two con-
stants, consider the following 2-step rational Darboux transformation:
T1 = B1A1 + λ1,
T̃ = A1B1 + λ1 = A2B2 + λ2,
T2 = B2A2 + λ2.
The first transformation at energy level λ1 maps T1 to T̃ and is state-deleting, while the second
transformation at energy level λ2 maps T̃ to T2 and is state-adding (or equivalently, the inverse
transformation from T2 to T̃ is state-deleting). The confluent version arises when λ1 = λ2, i.e.,
we use seed functions at each of the two steps which are (at least formally) eigenfunctions of
the corresponding (formal) operator with the same eigenvalue. A full discussion of confluent
Darboux transformations from this point of view can be seen, for instance, in [28]. We can make
this notion more precise in the following definition.
Definition 2.1. Let T1, T2 be second-order operators with rational coefficients. We will say
that T1 and T2 are related by a rational confluent Darboux transformation if there exist first-
order operators A1, A2, B1, B2, all with rational coefficients, and a constant λ such that
A1B1 = A2B2, T1 = B1A1 + λ, T2 = B2A2 + λ.
Given polynomials τ(z), φ(z), we define the rational operators
A(τ, φ) := τ−1 (φDz − φz) ,
B(φ, τ) := A(φ, τ) ◦
(
1− z2
)
= φ−1
((
1− z2
)
(τDz − τz)− 2zτ
)
. (2.1)
The form of these operators coincides with the general form of the first-order operators appearing
in factorization of operators given in [7, Proposition 3.5], with a particular choice that ensures
that operator T̂ (τ) = B(φ, τ)A(τ, φ) is in the natural gauge [7, Definition 5.1]. In the proofs,
we will use the fact that, for a given function f , we have
A(τ, φ)f = τ−1 Wr(φ, f),
where Wr denotes the Wronskian determinant.
Throughout this section, we consider an exceptional Legendre operator T̂ (τ) with polynomials
{πi(z)}i∈N0 that satisfy
T̂ (τ)πi = λiπi, λi1 6= λi2 if i1 6= i2, i, i1, i2 ∈ N0. (2.2)
Exceptional Legendre Polynomials and Confluent Darboux Transformations 7
Our goal is to apply a rational CDT on this operator. To this end, for m ∈ N0 and t ∈ R, let us
define the following objects:
ρi1i2(z) :=
∫ z
−1
πi1(u)πi2(u)
τ(u)2
du, i1, i2 ∈ N0, (2.3)
τm(z; t) := τ(z) (1 + tρmm(z)) , (2.4)
πm;i(z; t) := (1 + tρmm(z))πi(z)− tρim(z)πm(z), i ∈ N0. (2.5)
For a lighter notation, we may omit the t dependence and write τm instead of τm(z; t). Note
that τ might already depend on a number of real parameters, so τm will depend on the same
parameters as τ , plus an extra parameter t.
Remark 2.2. In the rest of this Section, i.e., for the following four Propositions, we shall
assume that ρi1i2(z) defined by (2.3) is a rational function that vanishes at z = −1 and τm(z)
and πm;i(z, t) defined by (2.4)–(2.5) are polynomials in z.
If we start from an exceptional Legendre operator (1.1) for a given τ polynomial with eigen-
polynomials πi, these assumptions are far from obvious by looking at (2.3)–(2.5). In the next
section we will see that the assumptions hold whenever (2.2) does, i.e., that the rational CDT
between exceptional Legendre families is well defined.
Proposition 2.3. For m ∈ N0, let ρmm(z) and τm(z) be defined by (2.3)–(2.4) and satisfy the
assumptions of Remark 2.2. Then, T̂ (τ) and T̂ (τm) are related by a rational confluent Darboux
transformation with
A(τ, πm)B(πm, τ) = A(τm, πm)B(πm, τm),
T̂ (τ) = B(πm, τ)A(τ, πm) + λm,
T̂ (τm) = B(πm, τm)A(τm, πm) + λm.
Proof. The results follow from direct calculation with the previous definitions. �
The following lemma examines the behaviour at the endpoint z = −1 of a combination of
these objects, and it will be necessary to prove some of the following propositions.
Lemma 2.4. Let {πi}i∈N0 be polynomials that satisfy the eigenvalue equation (2.2) with (1.1)
and let ρi1,i2(z) be the rational functions defined by (2.3) that vanish at z = −1. Then,
(
1− z2
)Wr(πi, πm)
τ2
∣∣∣∣
z=−1
= 0.
Proof. Since ρim(z) is rational and vanishes at z = −1, we can write for given α, β ∈ N0
ρim(z) = (1 + z)1+αq(z), q(−1) = Cq 6= 0, α > 0,
τ(z) = (1 + z)βp(z), p(−1) = Cp 6= 0,
where q is a rational function and p is a polynomial. Thus
ρ′im(−1) =
πiπm
τ2
∣∣∣
z=−1
=
{
Cq if α = 1,
0 if α ≥ 2.
Using the eigenvalue equation we have
(
1− z2
)
(π′′i πm − πiπ′′m)− 2
((
1− z2
)τ ′
τ
+ z
)
(π′iπm − πiπ′m) = (λi − λm)πiπm.
8 M.Á. Garćıa-Ferrero, D. Gómez-Ullate and R. Milson
Evaluating both sides at z = −1, we have
Wr(πi, πm)
τ2
∣∣∣∣
z=−1
=
(λi − λm)
2(1− 2β)
πiπm
τ2
∣∣∣∣
z=−1
<∞.
Since the previous expression is bounded, the desired result is proved. �
The next proposition shows how to build the eigenpolynomials of the transformed operator,
by the use of the second order intertwining relations for CDTs
(B2A1)T1 = T2(B2A1), T1(B1A2) = (B1A2)T2.
Proposition 2.5. For m ∈ N0, let ρim(z), τm(z) and πm;i(z, t) be defined by (2.3)–(2.5) and
satisfy the assumptions of Remark 2.2.
Then, for i ∈ N0, we have
(λm − λi)ρim =
(
1− z2
)
τ−1A(τ, πm)πi, (2.6)
(λm − λi)πm;i = B(πm, τm)A(τ, πm)πi, (2.7)
T̂ (τm)πm;i = λiπm;i. (2.8)
Proof. We start by noticing that
(λm − λi)ρ′im = (λm − λi)
πiπm
τ2
=
((
1− z2
)
τ−1A(τ, πm)πi
)′
,
where for the last equality we use the eigenvalue equation (2.2), or equivalently the Sturm–
Liouville equation((
1− z2
)
τ−2π′i
)′
+
(
1− z2
)
τzzτ
−3πi = λiτ
−2πi.
The first result follows by integration since Lemma 2.4 ensures that(
1− z2
)
τ−1A(τ, πm)πi
∣∣
z=−1 =
(
1− z2
)Wr(πm, πi)
τ2
∣∣∣∣
z=−1
= 0.
The second identity follows by direct calculation using the definitions and previous identities.
Indeed, using (2.1) and (2.6) we have
B(πm, τm)A(τ, πm)πi = A(πm, τm) ◦
(
1− z2
)((λm − λi)ρim(
1− z2
)
τ−1
)
= (λm − λi)π−1m Wr(τm, τρim)
= (λm − λi)π−1m τ2 Wr(1 + tρmm, ρim)
and deriving ρim in (2.3) and using (2.5) leads to the desired result (2.7).
The third identity (2.8) follows trivially from (2.7) and the intertwining relation
T̂ (τm)B(πm, τm)A(τ, πm) = B(πm, τm)A(τ, πm)T̂ (τ). �
The next result derives the transformation rule for ρi1i2 under a CDT, and it is key to obtain
the norming constants of the transformed polynomials in Proposition 2.7.
Proposition 2.6. For m, i1, i2 ∈ N0, let ρi1i2(z), τm(z) and πm;i(z, t) be defined by (2.3)–(2.5)
and satisfy the assumptions of Remark 2.2. Then,∫ z
−1
πm;i1(u; t)πm;i2(u; t)
τm(u)2
du = ρi1i2(z)− tρi1m(z)ρi2m(z)
1 + tρmm(z)
, i1, i2 ∈ N0. (2.9)
Exceptional Legendre Polynomials and Confluent Darboux Transformations 9
Proof. The identity between the derivatives of both sides can be easily proved by direct
computation using the definitions. The desired result follows then by integration since
ρi1i2(−1) = 0. �
The last proposition of this section shows that the CDT of an exceptional Legendre family
falls into the same class under suitable bound on the introduced parameter t.
Proposition 2.7. Assume that τ does not vanish in [−1, 1] and that {πi}i∈N0 are exceptional
Legendre polynomials with∫ 1
−1
πi1(u)πi2(u)
τ(u)2
du = νi1δi1i2 ,
for constants νi > 0, i ∈ N0 and completeness in L2
(
[−1, 1], τ−2dz
)
. Let m ∈ N0 and set
νm;i :=
{
νi if i 6= m,
(t+ ν−1m )−1 if i = m.
Then, τm(z) > 0 on [−1, 1] if and only if νm;m > 0. In that case, the set {πm;i(z; t)}i∈N0 is
a family of exceptional Legendre polynomials with∫ 1
−1
πm;i1(u)πm;i2(u)
τm(u)2
du = νm;i1δi1i2 , i1, i2,m ∈ N0, (2.10)
and completeness in L2
(
[−1, 1], τ−2m dz
)
.
Proof. First, note that (2.10) is true in a formal sense. By (2.9), the rational function
ρm;i1i2(z; t) := ρi1i2(z)− tρi1m(z)ρi2m(z)
1 + tρmm(z)
is defined by the integral on the l.h.s. of (2.9). Furthermore, since we are assuming that
ρi1i2(1) = δi1i2νi1 ,
we have
ρm;i1i2(1; t) = δi1i2
(
νi1 − δi1m
tν2m
1 + tνm
)
= δi1i2νm;i1 .
By (2.4), τm(z) is positive on z ∈ [−1, 1] if and only if the same is true for 1+ tρmm(z). Since
ρmm(z) is an increasing function, the latter is true if and only if 1 + tνm > 0. Observe that
ν−1m;m = t+ ν−1m = ν−1m (1 + tνm).
Hence τm(z) is positive on z ∈ [−1, 1] if and only if νm;m > 0.
Finally, we prove completeness. We assume that the eigenpolynomials {πi(z)}i∈N0 are L2-
complete in [−1, 1] relative to τ(z)−2dz. Following an argument adapted from the appendix
of [1], we re-express the completeness assumption as∑
i∈N0
ν−1i
πi(z)
τ(z)
πi(w)
τ(w)
= δ(z − w),
where the equality is understood in distributional sense on [−1, 1]× [−1, 1]. Rewriting (2.3) as
ρi1i2(z) =
∫ 1
−1
θ(z − u)
πi1(u)πi2(u)
τ(u)2
du, i1, i2 ∈ N0,
10 M.Á. Garćıa-Ferrero, D. Gómez-Ullate and R. Milson
where θ(z) denotes the Heaviside step function, it follows that for j ∈ N0∑
i∈N0
ν−1i
πi(z)
τ(z)
ρij(w) = θ(w − z)πj(z)
τ(z)
,
∑
i∈N0
ν−1i ρij(z)ρij(w) = θ(w − z)ρjj(z) + θ(z − w)ρjj(w).
By (2.4) and (2.5), we have
πm;i(z)
τm(z)
=
πi(z)
τ(z)
− tρim(z)
πm(z)
τm(z)
, i ∈ N0,
tρmm(z)
τm(z)
=
1
τ(z)
− 1
τm(z)
.
Therefore, making use of the previous identities,∑
i∈N0
ν−1m;i
πm;i(z)
τm(z)
πm;i(w)
τm(w)
== t
πm;m(z)
τm(z)
πm;m(w)
τm(w)
+
∑
i∈N0
ν−1i
πm;i(z)
τm(z)
πm;i(w)
τm(w)
= t
πm(z)
τm(z)
πm(w)
τm(w)
+
∑
i∈N0
ν−1i
(
πi(z)
τ(z)
− tρim(z)
πm(z)
τm(z)
)(
πi(w)
τ(w)
− tρim(w)
πm(w)
τm(w)
)
= t
πm(z)
τm(z)
πm(w)
τm(w)
+ δ(z − w)− t θ(w − z)πm(z)
τ(z)
πm(w)
τm(w)
− t θ(z − w)
πm(z)
τm(z)
πm(w)
τ(w)
+ t2
(
θ(w − z)ρmm(z) + θ(z − w)ρmm(w)
)πm(z)
τm(z)
πm(w)
τm(w)
= δ(z − w). �
3 Recursive construction and proof of theorems
The strategy to prove the main theorems is the following. First, we will define some polynomials
and rational functions recursively, starting the recursion at the objects corresponding to the
classical Legendre Sturm–Liouville problem. The recursion formulas coincide with (1.6), (2.5)
and (2.9). Next, we show in Proposition 3.2 that these recursively defined objects coincide with
those defined in Definition 1.4, and thus they satisfy the rationality and polynomiality conditions
of Remark 2.2. Propositions 2.3–2.7 then ensure that at each step of the recursion we have an
exceptional Legendre Sturm–Liouville problem, provided the parameters are chosen in the right
range.
Definition 3.1. Let i, i1, i2 ∈ N0, m = mn = (m1, . . . ,mn) ∈ Nn0 and mj = (m1, . . . ,mj), and
define recursively functions Rm;i1i2(z; tm), τ̃m(z; tm) and P̃m;i(z; tm). For j = 0, we start the
recursion at m0 = ∅ with
Rm0;i1i2(z; tm0) = Ri1i2(z),
τ̃m0(z; tm0) = 1,
P̃m0;i(z; tm0) = Pi(z),
where Ri1i2(z) are given by (1.5) and Pi(z) are the classical Legendre operators. For j = 1, . . . , n
we define recursively
Rmj ;i1i2(z; tmj ) = Rmj−1;i1i2(z; tmj−1)
−
tmjRmj−1;i1mj (z; tmj−1)Rmj−1;i2mj (z; tmj−1)
1 + tmjRmj−1;mjmj (z; tmj−1)
, (3.1)
Exceptional Legendre Polynomials and Confluent Darboux Transformations 11
τ̃mj (z; tmj ) =
(
1 + tmjRmj−1;mjmj (z; tmj−1)
)
τ̃mj−1(z; tmj−1), (3.2)
P̃mj ;i(z; tmj ) = (1 + tmjRmj−1;mjmj (z; tmj−1))P̃mj−1;i(z; tmj−1)
− tmjRmj−1;imj (z; tmj−1)P̃mj−1;mj (z; tmj−1). (3.3)
The next proposition states that these recursively defined functions coincide with the poly-
nomials and rational functions introduced in Definition 1.4.
Proposition 3.2. For i, i1, i2 ∈ N0 and m = (m1, . . . ,mn) ∈ Nn0 , let Rm;i1i2(z; tm), τ̃m(z; tm)
and P̃m;i(z; tm) be the functions defined in Definition 3.1. Let τm(z; tm) and Pm;i(z; tm) be the
polynomials defined in Definition 1.4. Then,
τ̃m(z; tm) = τm(z; tm), (3.4)
P̃m;i(z; tm) = Pm;i(z; tm), (3.5)
and
Rm;i1i2(z; tm) =
∫ z
−1
Pm;i1(u; tm)Pm;i2(u; tm)
τm(u; tm)2
du, (3.6)
where, again, the integral denotes an anti-derivative that vanishes at z = −1.
The consequence of this proposition is to ensure that the rational CDTs applied iteratively
on the classical Legendre operator are always well defined, i.e., the conditions specified in Re-
mark 2.2 will hold at each step of the chain. Before we can address the proof of Proposition 3.2,
we need to establish the following technical lemma.
Lemma 3.3. Let m ∈ Nn0 with n ≥ 3. Then(
1 + tmn−1Rmn−2;mn−1mn−1(z; tmn−2) tmnRmn−2;mn−1mn(z; tmn−2)
tmn−1Rmn−2;mn−1mn(z; tmn−2) 1 + tmnRmn−2;mnmn;(z; tmn−2)
)−1
=Mn−1,n
(
Rm(z; tm)−1
)
, (3.7)
where Mn−1,n denotes the bottom right 2× 2 submatrix of the indicated matrix and Rm(z; tm)
is given in Definition 1.4.
Proof. The result follows by iteration and the Sylvester determinant identity [2]. Let us start
by showing the argument for n = 3. If we write
Rm(z; tm) =
a b1
bT2 A
,
where bj are 2-tuples for j = 1, 2 and A is the M2,3 submatrix, by the Sylvester determinant
identity,
M2,3
(
Rm(z; tm)−1
)
=
(
A− bT2 a
−1b1
)−1
.
Identifying the elements in the decomposition of Rm(z; tm) according to its definition, by direct
calculation and the recursive formulation (3.1) we obtain
A− b1a
−1bT2 =
(
1 + tm2Rm1;m2m2(z; tm1) tm3Rm1;m2m3(z; tm1)
tm2Rm1;m2m3(z; tm1) 1 + tm3Rm1;m3m3(z; tm1)
)
as desired.
12 M.Á. Garćıa-Ferrero, D. Gómez-Ullate and R. Milson
For n > 3, let R[j]
m(z; tm) be the (n− j)× (n− j) matrix for any j ∈ {1, n− 2} with entries[
R[j]
m(z; tm)
]
k`
:= δk` + tmj+`
Rmj ;mj+kmj+`
(z; tmj ), k, ` ∈ {1, . . . , n− j}. (3.8)
We aim to show that R[n−2]
m (z; tm)−1 = Mn−1,n
(
Rm(z; tm)−1
)
. Arguing as above, we have
R[n−2]
m (z; tm)−1 = M2,3
(
R[n−3]
m (z; tm)−1
)
. Applying analogously the Sylvester determinant
identity, we also obtain
R[j+1]
m (z; tm)−1 =M2,n−j
(
R[j]
m(z; tm)−1
)
for j ∈ {0, . . . , n− 3}. The result then follows by iteration. �
Proof of Proposition 3.2. Throughout this proof, we are going to omit the explicit depen-
dence on z and tm of the objects, which must be understood from the dependence on m, i.e.,
we will write Rm instead of Rm(z; tm). By definition, we have
Rm =
Rmn−1
...
· · · 1 + tmnRmnmn
.
Applying the expression for the inverse with the adjoint of the cofactors matrix, we obtain that[
R−1m
]
nn
=
detRmn−1
detRmn
=
τmn−1
τmn
.
Applying Lemma 3.3 after computing the inverse of the matrix in the left hand side of (3.7)
and using the recursion (3.1) allows to prove that the τm and τmn−1 satisfy the recursion
relation (3.2). Since τ̃m1 = τm1 , we see that (3.4) holds.
In order to prove (3.5), we first observe that (3.3) can be rewritten as
P̃m;i =
τ(m;i)
τmn−1
[(
R[n−1]
(m,i)
)−1 (
P̃mn−1;mn , P̃mn−1;i
)T]
1
,
where R[n−1]
(m,i) = R[n−1]
(m,i)(z; t(m,i)) is given by (3.8). The proof follows by induction. It is clear
that (3.5) holds for m1 by definition. We assume that it also holds for mj with j = 1, . . . , n−1,
and we must prove it also hods for mn. We start by proving that,
(
R[n−1]
(m,i)
)−1(Pmn−1;mn
Pmn−1;i
)
=
τmn−1
τmn−2
(R[n−2]
(m,i)
)−1Pmn−2;mn−1
Pmn−2;mn
Pmn−2;i
〈1〉 , (3.9)
where in the right hand side we have the last two components of a vector of three entries. This
may be verified as follows. First note that, by assumption, we have P̃mj ;i = Pmj ;i for all i ∈ N0
and j = 1, . . . , n− 1. By Lemma 3.3 and (3.3), the left hand side is equal to
M2,3
((
R[n−2]
(m,i)
)−1)−tmn−1Rmn−2;mn−1,mn
τmn−1
τmn−2
0
−tmn−1Rmn−2;mn−1,i) 0
τmn−1
τmn−2
Pmn−2;mn−1
Pmn−2;mn
Pmn−2;i
,
where we have used by (3.2)
1 + tmn−1Rmn−2;mn−1,mn−1 =
τmn−1
τmn−2
.
Exceptional Legendre Polynomials and Confluent Darboux Transformations 13
Now we can conclude that
M2,3
((
R[n−2]
(m,i)
)−1)−tmn−1Rmn−2;mn−1,mn
τmn−1
τmn−2
0
−tmn−1Rmn−2;mn−1,i 0
τmn−1
τmn−2
(3.10)
corresponds to the last two rows of
(
R[n−2]
(m,i)
)−1
multiplied by
τmn−1
τmn−2
. For the second block,
corresponding to M2,3
((
R[n−2]
(m,i)
)−1)
, the correspondence is clear. In order to verify the result
for the elements (2, 1) and (3, 1) of
(
R[n−2]
(m,i)
)−1
, we identify the components of the second matrix
in (3.10) as elements of R[n−2]
(m,i) and we rely on the fact that we are multiplying elements of the
matrix R[n−2]
(m,i) and its inverse. In fact, for instance the element (2, 1) corresponds to
−
[(
Rn−2(m,i)
)−1]
2,2
[
R[n−2]
(m,i)
]
2,1
−
[(
R[n−2]
(m,i)
)−1]
2,3
[
R[n−2]
(m,i)
]
3,1
= −
[
I
]
2,1
+
[(
R[n−2]
(m,i)
)−1]
2,1
[
R[n−2]
(m,i)
]
1,1
=
(
1 + tmn−1Rmn−2;mn−1mn−1
) [(
R[n−2]
(m,i)
)−1]
2,1
=
τmn−1
τmn−2
[(
R[n−2]
(m,i)
)−1]
2,1
,
where I is the identity matrix. Similarly to (3.9), we can show the following identity for
j = 1, . . . , n− 2:
(
R[j]
(m,i)
)−1
Pmj ;mj+1
...
Pmj ;mn
Pmj ;i
=
τmj
τmj−1
(R[j−1]
(m,i)
)−1
Pmj−1;mj
...
Pmj−1;mn
Pmj−1;i
〈1〉
.
Combining the previous identities yields (3.5):
P̃m;i =
τ(m;i)
τmn−1
[(
R[n−1]
(m,i)
)−1 (
P̃mn−1;mn , P̃mn−1;i
)T]
1
,
=
τ(m;i)
τmn−2
[(
R[n−2]
(m,i)
)−1 (
Pmn−2;mn−1 , Pmn−2;mn , Pmn−2;i
)T]
1
= · · ·
= τ(m;i)
[(
R(m,i)
)−1 (
Pm1 , . . . Pmn , Pi
)T]
1
= Pm;i.
Finally, relation (3.6) follows by Proposition 2.6 and by induction on j. �
Proposition 3.2 together with the results in Section 2 allow now to prove the main theorems.
Proof of Theorems 1.5 and 1.6. The key to the proof is to observe that starting form the
classical Legendre operator, the application of a rational confluent Darboux transformation in-
dexed by an integer mi introduces an extra real parameter tmi and leads to a well defined
Sturm–Liouville problem defining a family of exceptional Legendre polynomials. Indeed, the
equivalence of the objects defined by the CDT recursion (2.3)–(2.5) and those defined by ma-
trix multiplication in Definition 1.4 show that for any m ∈ Nn0 , τm(z; tm) and Pm;i(z; tm) are
polynomials andRm(z; tm) is a matrix of rational functions that satisfy the premises of Proposi-
tions 2.3–2.7 (see Remark 2.2). Theorem 1.5 follows then from Proposition 2.5 and Theorem 1.6
follows by induction from Proposition 2.7, which establishes the bounds on the parameters (1.10)
that ensure the regularity of Tm(tm) and the positivity of the measure in [−1, 1] �
14 M.Á. Garćıa-Ferrero, D. Gómez-Ullate and R. Milson
As mentioned above, the degree of the i-th exceptional Legendre polynomial Pm;i indexed
by m = (m1, . . . ,mn) ∈ Nn0 is not i. The next proposition provides this result.
Proposition 3.4. Let m = (m1, . . .,mn) and suppose that m1, . . .,mn are distinct. Let τm, Pm;i
be as defined in (1.6) and (1.8). Then,
deg τm(z; tm) = 2(m1 + . . .+mn) + n, (3.11)
degPm;i(z; tm) = 2(m1 + . . .+mn) + n+ i
− (δi,m1 + · · ·+ δi,mn)(2i+ 1), i ∈ N0. (3.12)
Moreover,
Pm;mk
(z; tm) = Pm〈k〉;mk
(z; tm〈k〉), k = 1, . . . , n, (3.13)
where m〈k〉 ∈ Nn−10 denotes the tuple obtained by removing the kth entry of m.
Notice that (3.13) accounts for the above Kronecker delta terms in (3.12). It is also worth
noting that, as opposed to the “traditional” exceptional families, the degree sequence for ex-
ceptional Legendre polynomials is not an increasing sequence, which is further evidence of the
different construction.
Proof. The identity (3.13) follows directly from applying (3.3) to this specific choice and the
symmetry with respect to permutations in m.
Relations (3.11) and (3.12) can be proved by induction. It is clear that they hold for m1 = m1,
since degRi1i2(z) = i1 + i2 + 1. Notice that in Pm1;i(z; tm1), the coefficients of Rm1m1(z)Pi(z)
and Rm1i(z)Pm1(z) do not coincide if i 6= m1, so degPm1;i(z; tm1) = 2m1 + i + 1. If i = m1,
by (3.13) we obtain degPm1;m1(z; tm1) = degPm1(z) = m1
Now we assume (3.11) and (3.12) hold for mj−1. By (3.6), the degree of Rmj−1;imj (z; tmj−1),
understood as the difference between the degree of the polynomial numerator and of the poly-
nomial denominator, is mj + i+ 1 if i 6= m1, . . . ,mj−1. Then,
deg τmj (z; tmj ) = deg τmj−1(z; tmj−1) + 2mj + 1 = 2(m1 + · · ·+mj) + j.
Arguing as above to verify that there is no cancellation between the highest order contributions,
we obtain
degPmj ;i(z; tmj ) = 2(m1 + · · ·+mj−1) + (j − 1) + 2mj + i+ 1
if i 6= m1, . . . ,mj . In order to prove the result for i equal to some component of mj , we employ
relation (3.13). �
Remark 3.5. From Proposition 3.4 we see that the codimension (number of missing degrees)
of the exceptional Legendre family indexed by m = (m1, . . . , n) is 2(m1 + · · · + mn) + n. This
coincides with the degree of τm, as it happens for all exceptional polynomials [7].
So far we have considered the case when m = (m1, . . . ,mn) contains distinct indices. As
announced in Remark (1.7), let us show that this choice entails no loss of generality.
Proposition 3.6. Let m ∈ Nn0 and let τm(z; tm) and Pm;i(z; tm) be as defined in (1.6)
and (1.8). Then, for any j ∈ N0 we have
τ(m,j,j)
(
z; (tm, tj , tj)
)
= τ(m,j)
(
z; (tm, 2tj)
)
,
P(m,j,j);i
(
z; (tm, tj , tj)
)
= P(m,j);i
(
z; (tm, 2tj)
)
.
Exceptional Legendre Polynomials and Confluent Darboux Transformations 15
We see thus that the repeated application of a 2-step confluent Darboux transformation at
the same eigenvalue only serves to modify the deformation parameter. In general, if the two
parameters at the repeated j are different, we would have similarly
τ(m,j,j)(z; (tm, tj , t
′
j) = τ(m,j)(z; (tm, tj + t′j))),
P(m,j,j);i(z; (tm, tj , t
′
j)) = P(m,j);i(z; (tm, tj + t′j).
Proof. We omit again explicit dependence on z and tm, tj if no confusion arises. We apply
Proposition 3.2 and Definition 3.1 twice to obtain
τ(m,j,j) = (1 + tjR(m,j);jj)τ(m,j) =
(
1 + tjRm;jj −
t2jR
2
m;jj
1 + tjRm;jj
)
(1 + tjRm;jj)τm
= (1 + 2tjRm;jj)τm = τ(m,j)
(
(z; (tm, 2tj)
)
,
P(m,j,j);i = (1 + tjR(m,j);jj)P(m,j);i − tjR(m,j);ijP(m,j);j
= (1 + tjR(m,j);jj)
(
(1 + tjRm;jj)Pm;i − tjRm;ijPm;i
)
− tj
(
Rm;ij −
tjRm;jjRm;ij
1 + tjRm;jj
)
Pm;j
= (1 + 2tjRm;jj)Pm;i − 2tjRm;ijPm;j = P(m,j);i(z; (tm, 2tj)). �
4 Examples
To conclude, we present some examples of exceptional Legendre polynomials and orthogonality
relations for the cases of n = 1 and n = 2.
4.1 The 1-parameter exceptional Legendre family
In the 1-parameter case, we have m = (m1) ∈ N0, and
τm1(z; tm1) = 1 + tm1Rm1m1(z),
Pm1;i(z; tm1) = (1 + tm1Rm1m1(z))Pi(z)− tm1Rim1(z)Pm1(z), i ∈ N0.
Note that Pm1;m1(z) = Pm1(z). The degrees of the other polynomials are
degPm1;i(z) = i+ 2m1 + 1, i 6= m1.
The corresponding exceptional operator is Tm1(tm1) = T̂ (τm1(z; tm1)) with the latter as per (1.1).
The polynomial τm1(z; tm1) does not vanish in [−1, 1] provided that
tm1 > −
1
Rm1m1(1)
= −m1 −
1
2
.
In this case, {Pm1;i(z; tm1)}i∈N0 is a family of exceptional Legendre polynomials, with orthogo-
nality weight
Wm1(z; tm1) =
1
τm1(z; tm1)2
.
The above set is a complete orthogonal polynomial basis of the space L2([−1, 1],Wm1(z; tm1)dz).
The orthogonality relations are∫ 1
−1
Pm1;i1(u; tm1)Pm1;i2(u; tm1)
τm1(u; tm1)2
du =
2
1 + 2i1
δi1i2 , i1, i2 ∈ N0\{m1},∫ 1
−1
Pm1;m1(u; tm1)2
τm1(u; tm1)2
du =
2
1 + 2m1 + 2tm1
.
16 M.Á. Garćıa-Ferrero, D. Gómez-Ullate and R. Milson
-1.0 -0.5 0.0 0.5 1.0
0.2
0.4
0.6
0.8
1.0
Figure 1. Exceptional Legendre weight τ−2
m1
(z, tm1
) for m1 = 4 and tm1
= 5.2.
The above example illustrates perfectly the isospectral nature of the CDT that relates T (1)→
Tm1(tm1). The eigenvalues of the two operators are the same. As for the eigenfunctions, if
tm1 6= 0, then for i 6= m1 they are transformed, but their norms stay the same. On the other
hand, for i = m1 the opposite happens: the eigenfunction does not change but its norm does.
Since τ ′m1
(z) = tm1P
2
m1
(z), it follows that the weight Wm1(z) is a decreasing (resp. increasing)
function for tm1 > 0 (resp. tm1 < 0), which has m1 saddle points in [−1, 1] at the zeros of Pm1 ,
but no local minima or maxima.
For instance, for m1 = 4, we have
τ4(z, t4) = 1 +
1
576
t4
(
64 + 81z − 540z3 + 1998z5 − 2700z7 + 1225z9
)
and the weight is shown in Fig. 1. The first few polynomials for this choice are
P4;0(z, t4) = 1 +
t4
144
(
16 + 135z3 − 459z5 + 585z7 − 245z9
)
,
P4;1(z, t4) = z +
t4
1152
(
− 9 + 128z + 171z2 + 30z4 − 1314z6 + 2475z8 − 1225z10
)
,
P4;2(z, t4) = P2(z)−
t4
576
(
32− 96z2 + 189z3 − 756z5 + 1278z7 − 1300z9 + 525z11
)
,
P4;3(z, t4) = P3(z)−
t4
9216
(
243− 1536z − 3402z2 + 2560z3 + 3645z4 + 7668z6 − 17955z8
+ 16950z10 − 6125z12
)
,
P4;4(z, t4) = P4(z),
P4;5(z, t4) = P5(z) +
t4
9216
(
243 + 1920z − 1215z2 − 8960z3 − 3645z4 + 8064z5 + 17145z6
− 42255z8 + 66171z10 − 50855z12 + 15435z14
)
.
We display the above polynomials for t4 = 0 (classical Legendre) and for t4 = 5.2 in Fig. 2. Ob-
serve that all polynomials undergo a continuous deformation as t4 changes, except for P4;4 = P4
that stays the same.
4.2 The 2-parameter exceptional Legendre family
In the 2-parameter case, we have m = (m1,m2) ∈ N2
0, tm = (tm1 , tm2) and
τm(z; tm) = τm1(z; tm1)τm2(z; tm2)− tm1tm2Rm1m2(z)2,
Pm;i(z; tm) = Pi(z)τm(z; tm)− Pm1(z)tm1τm2(z; tm2)Rm1;m2,i(z; tm2)
− Pm2(z)tm2τm1(z; tm1)Rm1;m2,i(z; tm1), i ∈ N0,
Exceptional Legendre Polynomials and Confluent Darboux Transformations 17
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
1.5
2.0
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
1.5
2.0
Figure 2. First few exceptional Legendre polynomials Pm1;i(z; tm1) for m1 = 4, with tm1 = 0 (left) and
tm1
= 5.2 (right).
where
Rm1;i1i2(z; tm) =
∫ z
−1
Pm1;i1(u; tm1)Pm1;i2(u; tm1)
τm1(u; tm1)2
du
= Ri1i2(z)− tm1Ri1m1(z)Ri2m1(z)
1 + tm1Rm1m1
, i1, i2 ∈ N0,
Supposing that m1 6= m2 we have
degPm;i(z; tm) = i+ 2m1 + 2m2 + 2, i ∈ N0\{m1,m2},
degPm,m1(z; tm) = m1 + 2m2 + 1,
degPm;m2(z; tm) = m2 + 2m1 + 1.
The polynomial τm(z; tm1,tm2
) does not vanish in [−1, 1] provided that
tm1 > −m1 −
1
2
, tm2 > −m2 −
1
2
.
In this case, {Pm;i(z; tm)}i∈N0 is a family of exceptional Legendre polynomials, with orthogo-
nality weight
Wm(z; tm) =
1
τm(z; tm)2
.
The above set is a complete orthogonal polynomial basis of the space L2([−1, 1],Wm(z; tm)dz).
The orthogonality relations are∫ 1
−1
Pm;i2(u; tm)Pm;i2(u; tm)
τm(u; tm)2
du =
2
1 + 2i1
δi1i2 , i1, i2 ∈ N0\{m1,m2},∫ 1
−1
Pm;m1(u; tm)2
τm(u; tm)2
du =
2
1 + 2m1 + 2tm1
,∫ 1
−1
Pm;m2(u; tm)2
τm(u; tm)2
du =
2
1 + 2m2 + 2tm2
.
For instance, for m = (m1,m1) = (1, 2), we have
τ(1,2)
(
z; (t1, t2)
)
= 1 +
1
3
t1
(
1 + z3
)
+
1
20
t2
(
4 + 5z − 10z3 + 9z5
)
+
1
960
t1t2(1 + z)4
(
49− 116z + 110z2 − 36z3 + 9z4
)
.
For certain values of tm = (tm1 , tm2) the weight is displayed in Fig. 3. To the best of our
knowledge, this is the first example of an exceptional orthogonal polynomial system whose
weight is not monotonic or unimodal.
18 M.Á. Garćıa-Ferrero, D. Gómez-Ullate and R. Milson
-1.0 -0.5 0.0 0.5 1.0
0.5
1.0
1.5
Figure 3. Exceptional Legendre weight τ−2
m (z, tm) for m = (m1,m2) = (1, 2) and (tm1
, tm2
) = (2,−1.6).
5 Summary
In this paper we show that the class of exceptional orthogonal polynomials is much larger than
previously thought. A new construction based on confluent Darboux transformations leads to
new exceptional families with some similarities and differences with respect to the other excep-
tional families. In common, they are also Sturm–Liouville problems with rational coefficients,
each family is indexed by a set of integers, and it defines a complete basis of polynomial eigen-
functions, whose degree sequence has missing degrees. But as opposed to the other exceptional
families, there are no gaps in the spectrum and the new families contain an arbitrary number
of real deformation parameters, so the construction can be seen as an isospectral deformation
of the classical operators. We illustrate the new construction by describing the full class of
exceptional Legendre polynomials, which cannot be derived through the standard construction.
The same method can be applied with minor modifications to the Jacobi operator. A more
exhaustive description of these matters will be provided in a forthcoming publication, together
with a discussion of the implications for the classification of exceptional polynomials [8].
Acknowledgements
MAGF would like to thank the Max-Planck-Institute for Mathematics in the Sciences, Leipzig
(Germany), where some of her work took place. DGU acknowledges support from the Spanish
MICINN under grants PGC2018-096504-B-C33 and RTI2018-100754-B-I00 and the European
Union under the 2014-2020 ERDF Operational Programme and by the Department of Economy,
Knowledge, Business and University of the Regional Government of Andalusia (project FEDER-
UCA18-108393).
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1 Introduction and main results
1.1 Notation and definitions
1.2 Main results
2 One step confluent Darboux transformations
3 Recursive construction and proof of theorems
4 Examples
4.1 The 1-parameter exceptional Legendre family
4.2 The 2-parameter exceptional Legendre family
5 Summary
References
|
| id | nasplib_isofts_kiev_ua-123456789-211172 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T07:43:27Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | García-Ferrero, María Ángeles Gómez-Ullate, David Milson, Robert 2025-12-25T13:21:37Z 2021 Exceptional Legendre Polynomials and Confluent Darboux Transformations. María Ángeles García-Ferrero, David Gómez-Ullate and Robert Milson. SIGMA 17 (2021), 016, 19 pages 1815-0659 2020 Mathematics Subject Classification: 33C47; 34L10; 34A05 arXiv:2008.02822 https://nasplib.isofts.kiev.ua/handle/123456789/211172 https://doi.org/10.3842/SIGMA.2021.016 Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of ''exceptional'' degrees. In this paper, we introduce a new construction of multi-parameter exceptional Legendre polynomials by considering the isospectral deformation of the classical Legendre operator. Using confluent Darboux transformations and a technique from inverse scattering theory, we obtain a fully explicit description of the operators and polynomials in question. The main novelty of the paper is the novel construction that allows for exceptional polynomial families with an arbitrary number of real parameters. MAGF would like to thank the Max-Planck-Institute for Mathematics in the Sciences, Leipzig (Germany), where some of her work took place. DGU acknowledges support from the Spanish MICINN under grants PGC2018-096504-B-C33 and RTI2018-100754-B-I00 and the European Union under the 2014-2020 ERDF Operational Programme and by the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia (project FEDERUCA18-108393). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Exceptional Legendre Polynomials and Confluent Darboux Transformations Article published earlier |
| spellingShingle | Exceptional Legendre Polynomials and Confluent Darboux Transformations García-Ferrero, María Ángeles Gómez-Ullate, David Milson, Robert |
| title | Exceptional Legendre Polynomials and Confluent Darboux Transformations |
| title_full | Exceptional Legendre Polynomials and Confluent Darboux Transformations |
| title_fullStr | Exceptional Legendre Polynomials and Confluent Darboux Transformations |
| title_full_unstemmed | Exceptional Legendre Polynomials and Confluent Darboux Transformations |
| title_short | Exceptional Legendre Polynomials and Confluent Darboux Transformations |
| title_sort | exceptional legendre polynomials and confluent darboux transformations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211172 |
| work_keys_str_mv | AT garciaferreromariaangeles exceptionallegendrepolynomialsandconfluentdarbouxtransformations AT gomezullatedavid exceptionallegendrepolynomialsandconfluentdarbouxtransformations AT milsonrobert exceptionallegendrepolynomialsandconfluentdarbouxtransformations |