Determinantal Formulas for Exceptional Orthogonal Polynomials
We present determinantal formulas for families of exceptional ₘ-Laguerre and exceptional ₘ-Jacobi polynomials and also for exceptional ₂-Hermite polynomials. The formulas resemble Vandermonde determinants and use the zeros of the classical orthogonal polynomials.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 047, 16 pages
Determinantal Formulas
for Exceptional Orthogonal Polynomials
Brian SIMANEK
Baylor University Math Department, Waco, Texas, 76706, USA
E-mail: Brian Simanek@baylor.edu
URL: https://www.baylor.edu/math/index.php?id=925780
Received February 11, 2022, in final form June 17, 2022; Published online June 25, 2022
https://doi.org/10.3842/SIGMA.2022.047
Abstract. We present determinantal formulas for families of exceptional Xm-Laguerre and
exceptional Xm-Jacobi polynomials and also for exceptional X2-Hermite polynomials. The
formulas resemble Vandermonde determinants and use the zeros of the classical orthogonal
polynomials.
Key words: exceptional orthogonal polynomials; determinantal formulas
2020 Mathematics Subject Classification: 42C05; 33C47
1 Introduction
The classical families of orthogonal polynomials known as Laguerre, Jacobi, and Hermite poly-
nomials have been studied extensively over the past 200 years because of their applicability
and convenient formulas. Many of their applications are derived from the fact that they are
polynomial solutions of explicit second order linear differential equations with polynomial coef-
ficients. Exceptional orthogonal polynomials arise in a similar context: as polynomial solutions
of explicit second order linear differential equations with rational coefficients. However, for the
exceptional orthogonal polynomial sequences, there are a finite number of natural numbers for
which there is no polynomial of that degree.
Exceptional orthogonal polynomials (XOPs) were first introduced in [16]. In [12], it was
shown that every sequence of XOPs is related to a classical orthogonal polynomial sequence by
a finite number of Darboux transformations. This allows us to label every sequence of XOPs
as either Laguerre, Jacobi, or Hermite. It also naturally leads us to wonder what properties
of the classical polynomials remain valid for the XOPs. For example, the classical orthogonal
polynomials are known to satisfy three-term recurrence relations and one could ask if there
is an analogous result for XOPs. Indeed, Durán showed in [5, 8] that exceptional orthogonal
polynomial sequences also satisfy recurrence relations, though of a different form (see also [17]).
The zeros of exceptional orthogonal polynomials present especially interesting phenomena
that have inspired considerable study. These zeros divide into two collections: the regular zeros
and the exceptional zeros. The regular zeros exhibit behavior that mimics the behavior of the
zeros of the classical orthogonal polynomials, while the exceptional zeros do not. Due to the
presence of these exceptional zeros, the zeros of the classical orthogonal polynomials are easier
to understand and study than the zeros of the XOPs. Detailed information about the zero
behavior of exceptional orthogonal polynomial sequences can be found in [1, 2, 18, 23, 29].
Our analysis will focus on determinantal formulas for certain families of XOPs. There are
ways to express the classical orthogonal polynomials using determinants of matrices whose entries
are moments of the measure of orthogonality. A comparable result for the most elementary
families of XOPs follows from [20, 21, 26]. Durán found alternative determinantal formulas for
mailto:Brian_Simanek@baylor.edu
https://www.baylor.edu/math/index.php?id=925780
https://doi.org/10.3842/SIGMA.2022.047
2 B. Simanek
XOPs in [3, 4, 6] by considering limits of discrete XOPs (see also [9]). Our main results will
be formulas for exceptional Laguerre, exceptional Jacobi, and exceptional Hermite polynomials
as determinants of matrices whose entries are powers of zeros of the corresponding classical
orthogonal polynomials (except in the last row, where the entries are explicit polynomials).
Consequently, the Vandermonde determinant of the zeros of the classical orthogonal polynomials
will appear as a factor in the resulting formulas.
In Section 2, we will review the basic facts and formulas about classical and exceptional
orthogonal polynomials that will be relevant for our later calculations. In Section 3, we will
derive our new determinantal formulas. In every case, we will first need to prove a first order
differential relation for the polynomials (some of which were already known). In the Hermite
case, our determinantal formula will only apply to one specific exceptional orthogonal polynomial
sequence. For the Laguerre and Jacobi cases, our results will apply to XOP sequences that are
obtained from the classical sequences by a single Darboux transformation.
2 Background and formulas
2.1 Classical orthogonal polynomials
The starting place for our calculations will be a closer examination of the relationship between
the exceptional orthogonal polynomials and the classical orthogonal polynomials. The next
several sections will briefly catalog some of the relevant formulas that will be needed in our
calculations. All of the formulas below involving the classical families of orthogonal polynomials
can be found in the Digital Library of Mathematical Functions [27].
2.1.1 Classical Laguerre polynomials
For any α > −1, the degree n Laguerre polynomial L
(α)
n (x) is given by
L(α)
n (x) =
n∑
m=0
(−1)m
m!
(
n+ α
n−m
)
xm. (2.1)
These polynomials are orthogonal in the space L2
(
[0,∞), xαe−xdx
)
. Notice that the leading
coefficient of L
(α)
n (x) is (−1)n/n! ̸= 0, so even if α ≤ −1, the formula (2.1) still defines a poly-
nomial of degree n.
There are many interesting and applicable identities involving Laguerre polynomials. We will
only need a small collection of these identities in our calculations. The formulas that will be
most relevant for us are the following:
d
dx
L(α)
n (x) = −L
(α+1)
n−1 (x), (2.2)
L(α)
n (x) = L(α+1)
n (x)− L
(α+1)
n−1 (x), (2.3)
xL(α+1)
n (x) = (n+ α+ 1)L(α)
n (x)− (n+ 1)L
(α)
n+1(x). (2.4)
Another relevant fact is that the polynomial L
(α)
n is the unique polynomial of degree n that
solves the ODE
xy′′ + (α+ 1− x)y′ + ny = 0
with normalization
L(α)
n (0) =
(
α+ n
n
)
.
We will also need the following result about the zeros of classical Laguerre polynomials.
Determinantal Formulas for Exceptional Orthogonal Polynomials 3
Theorem 2.1. Suppose L
(α)
n is defined by (2.1).
(i) If α > −1, then the zeros of L
(α)
n are simple and lie in the interval (0,∞).
(ii) If m ∈ N and α > m− 1, then the zeros of L
(−α−1)
m are simple.
(iii) If m ∈ N and α > m− 1, then L
(−α−1)
m does not have any zeros in (0,∞).
Part (i) of Theorem 2.1 is part of a well-known classical result. Part (ii) follows from [30,
equation (6.71.6)] and part (iii) follows from [30, Theorem 6.73]. Note that [30, Section 6.73]
states that [30, equation (6.71.6)] is valid for all α ∈ R.
2.1.2 Classical Jacobi polynomials
If α, β > −1, the degree n Jacobi polynomial P
(α,β)
n (x) is given by
P (α,β)
n (x) =
Γ(α+ n+ 1)
n!Γ(α+ β + n+ 1)
n∑
m=0
(
n
m
)
Γ(α+ β + n+m+ 1)
Γ(α+m+ 1)
(
x− 1
2
)m
. (2.5)
These polynomials are orthogonal in the space L2
(
[−1, 1], (1−x)α(1+x)βdx
)
. Even if α, β ≤ −1,
the formula (2.5) still defines a polynomial, though it may have degree smaller than n (see [18,
equation (89)]). It is stated in [30, equation (6.72.3)] that as long as α+β+n ̸∈ {−1,−2, . . . ,−n},
then P
(α,β)
n has degree n.
We will not need much in the way of additional formulas for classical Jacobi polynomials,
but we will need the following result about the zeros of classical Jacobi polynomials.
Theorem 2.2. Suppose P
(α,β)
n is defined by (2.5).
(i) If α, β > −1, then the zeros of P
(α,β)
n are simple and lie in the interval (−1, 1).
(ii) If α, β ̸∈ {−1,−2, . . . ,−n} and α+β ̸∈ {−1−n,−2−n, . . . ,−2n}, then the zeros of P
(α,β)
n
are simple.
(iii) If β, α+1−m ∈ (−1, 0) or β, α+1−m ∈ (0,∞), then P
(−α−1,β−1)
m has no zeros in (−1, 1).
Part (i) of Theorem 2.2 is part of a well-known classical result. Part (ii) follows from [30,
equation (6.71.5)] and part (iii) is stated in [18, Section 5.2]. Note that [30, Section 6.72] states
that [30, equation (6.71.5)] is valid for all α, β ∈ R.
2.1.3 Classical and generalized Hermite polynomials
The classical Hermite polynomials can be defined from the classical Laguerre polynomials by
the relations
H2n(x) = (−4)nn!L−1/2
n
(
x2
)
, H2n+1(x) = 2(−4)nn!xL1/2
n
(
x2
)
, (2.6)
from which it immediately follows that
d
dx
Hn(x) = 2nHn−1(x), n ≥ 1. (2.7)
We will also use the fact that
Hn(x) = 2xHn−1(x)−H ′
n−1(x). (2.8)
The relevant theorem about the zeros of Hermite polynomials is the fact that they are simple
and lie on the real line.
4 B. Simanek
To define the generalized Hermite polynomials, one first defines a partition λ = (λ1, . . . , λm)
∈ Nm, where
λ1 ≥ λ2 ≥ · · · ≥ λm ≥ 1.
The generalized Hermite polynomial Hλ is then defined by
Hλ = Wr[Hλm , Hλm−1+1, . . . ,Hλ2+m−2, Hλ1+m−1],
where Wr denotes the Wronskian determinant. It is known that the degree of Hλ is |λ| =
λ1 + · · · + λm (see [23, Section 2]). We call the partition λ an even partition if m is even and
λ2j = λ2j−1 for j = 1, . . . ,m/2. The relevant fact about the zeros of Hλ is the following result
from [23, Section 2].
Theorem 2.3. If λ is an even partition, then Hλ has no real zeros.
Simplicity of the zeros of Hλ is not known, but it was conjectured in [11] that the non-zero
zeros of Hλ are always simple (even if λ is not even).
2.2 Exceptional orthogonal polynomials
Now we are ready to shift our focus to the exceptional orthogonal polynomials. We will refer to
an XOP sequence with m missing degrees as an Xm polynomial sequence. These polynomials
have been heavily studied and many formulas are known that will be relevant to our calculations.
We will review these facts as well as some facts about the zeros of these polynomials.
2.2.1 Exceptional Xm-Laguerre polynomials
Exceptional Laguerre polynomials were originally introduced in [16] and have been studied in
papers such as [2, 4, 8, 10, 18, 19, 21, 25, 26, 28, 29]. We will focus on the Xm polynomials
that come from the application of a single Darboux transformation to the classical Laguerre
differential operator. These polynomials can be of type-I, type-II, or type-III (the more general
setting was discussed in [2]). While there is an extensive theory associated to these polynomials,
we will only discuss the portion that is relevant to our new results and refer the reader to the
aforementioned references for additional information.
In the type-I case, we require α > 0 and m ∈ N. The degree n polynomial in this sequence
is denoted L
I,(α)
m,n (x) and n can be any natural number strictly larger than m − 1 (that is,
polynomials of degree 0, 1, . . . ,m − 1 are omitted from this sequence). For such a choice of
parameters, it holds that
LI,(α)
m,n (x) = L(α)
m (−x)L
(α−1)
n−m (x) + L(α−1)
m (−x)L
(α)
n−m−1(x) (2.9)
(see [18, Proposition 3.1] or [25, equation (3.2)]). One can construct these polynomials using
partitions as in [2] by considering the empty partition and the singleton {m} (see [2, Proposi-
tion 4]).
In the type-II case, we require m ∈ N and α > m − 1. The degree n polynomial in this
sequence is denoted L
II,(α)
m,n (x) and n can be any natural number strictly larger than m− 1. For
such a choice of parameters, it holds that
LII,(α)
m,n (x) = xL(−α−1)
m (x)L
(α+2)
n−m−1(x) + (m− α− 1)L(−α−2)
m (x)L
(α+1)
n−m (x) (2.10)
(see [18, Proposition 4.1]). One can construct these polynomials using partitions as in [2] by
considering the empty partition and the m-tuple {1, 1, . . . , 1} (see [2, Proposition 4]).
Determinantal Formulas for Exceptional Orthogonal Polynomials 5
In the type-III case, we require α ∈ (−1, 0) and m ∈ N. The degree n polynomial in this
sequence is denoted L
III,(α)
m,n (x) and n can be 0 or any natural number strictly larger than m.
For such a choice of parameters and with n ≥ m+ 1, it holds that
LIII,(α)
m,n (x) = xL
(α+2)
n−m−2(x)L
(−α−1)
m (−x) + (m+ 1)L
(−α−2)
m+1 (−x)L
(α+1)
n−m−1(x) (2.11)
(see [25, equation (5.12)]). If n = 0, then L
III,(α)
m,0 (x) = 1. One can construct these polynomials
using partitions as in [2] by considering the m-tuple {1, 1, . . . , 1} and the empty partition (see
[2, Proposition 4]). The starting point for our investigation comes from the following formula,
which is [25, Theorem 5.2]:
LIII,(α)
m,n (x) = n
∫ x
0
L
(α+1)
n−m−1(t)L
(−α−1)
m (−t)dt+ (m+ 1)
(
n−m+ α
n−m− 1
)(
m− α− 1
m+ 1
)
, (2.12)
when n > m.
In the case m = 1, Kelly, Liaw and Osborn derived a determinantal formula for exceptional
Laguerre polynomials using what they call “adjusted moments” (see [20, Theorem 5.1]; see
also [26]). This was subsequently generalized to L
I,(α)
2,n in [21]. Durán [4] also discovered a
method for calculating exceptional Laguerre polynomials using determinantal formulas. Our
new results for exceptional Laguerre polynomials appear as Theorems 3.2, 3.4 and 3.5, all of
which are determinantal formulas involving the zeros of classical Laguerre polynomials. In the
type-I and type-II cases, we will need a result analogous to (2.12) (see Theorem 3.1), while in
the type-III case, we can appeal directly to (2.12).
2.2.2 Exceptional Xm-Jacobi polynomials
Exceptional Jacobi polynomials were originally introduced in [16] and have been studied in
papers such as [1, 6, 8, 13, 18, 19, 20, 24, 28, 29]. The polynomials we will consider come from
the application of a single Darboux transformation to the classical Jacobi differential operator
and are thus eigenfunctions of a second order differential operator (a more general setting was
discussed in [1]). They depend on a parameter m ∈ N and two real parameters α and β, so we
will denote the degree n polynomial by P
(α,β)
m,n and require n ≥ m (that is, polynomials of degree
0, 1, . . . ,m− 1 are omitted from this sequence). To ensure certain properties of the polynomial
degree and the zeros of these polynomials, we will insist that β > 0, that α+1−m > 0, and that
α+1−m−β ̸∈ {0, 1, . . . ,m−1}. In particular, the assumption α+1−m−β ̸∈ {0, 1, . . . ,m−1}
guarantees that the polynomial P
(−α−1,β−1)
m has degree m (see [30, equation (6.72.3)]). Further
justification for these assumptions is discussed in [18], but one can already see their importance
from Theorem 2.2. Under these assumptions, the following formula is valid:
P (α,β)
m,n (x) =
(−1)m
α+ 1 + n−m
(
(x− 1)(1 + α+ β + n−m)
2
P (−α−1,β−1)
m (x)P
(α+2,β)
n−m−1 (x)
+ (α+ 1−m)P (−α−2,β)
m (x)P
(α+1,β−1)
n−m (x)
)
, (2.13)
when n ≥ m (see [18, Section 5]). One can construct these polynomials using partitions as in [1]
by considering the empty partition and the m-tuple {1, 1, . . . , 1} (see [1, equation (5.4)]).
Durán [6] has found determinantal formulas for exceptional Jacobi polynomials and the afore-
mentioned results from [20] also apply to P
(α,β)
1,n . Our new results for these polynomials will in-
clude an integral formula comparable to (2.12) (see Theorem 3.6) and a determinantal formula
for the exceptional Jacobi polynomials that involves the zeros of the classical Jacobi polynomials
(see Theorem 3.7).
6 B. Simanek
2.2.3 Exceptional Xm-Hermite polynomials
Exceptional Hermite polynomials were originally introduced in [14] and have been studied in
papers such as [3, 5, 17, 23, 29]. The polynomials we will consider depend on a choice of parti-
tion λ as in Section 2.1.3. Given such a partition λ = (λ1, . . . , λm), one defines the exceptional
Hermite polynomial H
(λ)
|λ|,n by
H
(λ)
|λ|,n = Wr[Hλm , Hλm−1+1, . . . ,Hλ2+m−2, Hλ1+m−1, Hn−|λ|+m]
for n ≥ |λ| −m. Even for such n, H
(λ)
|λ|,n is identically zero unless n is in the set Nλ defined by
Nλ = {n ≥ |λ| −m : n ̸= |λ|+ λj − j, for j = 1, 2, . . . ,m}.
If n ∈ Nλ then the degree of H
(λ)
|λ|,n is n (see [23]).
Durán [3] has found determinantal formulas for exceptional Hermite polynomials. Our new
results for these polynomials will include an integral formula for H
({1,1})
2,n comparable to (2.12)
(see Theorem 3.8) and a determinantal formula for H
({1,1})
2,n that involves the zeros of the classical
Hermite polynomials (see Theorem 3.9). In this sense, our results are the natural extension of
[20, 21, 26] to the exceptional Hermite polynomials because we only consider the case when a
small number of degrees are missing from the sequence.
3 New formulas
In this section we will present both integral and determinantal formulas for various families of
exceptional orthogonal polynomials. All of the formulas and information about zeros that we
need to complete our calculations was provided in the previous section.
3.1 Xm Laguerre polynomials
We begin our analysis by stating the analog of (2.12) for the type-I and type-II exceptional
Xm-Laguerre polynomials.
Theorem 3.1. The following formulas are valid:
(i) If m ∈ N, α > 0, and n ≥ m, then
LI,(α)
m,n (x) =
α+ n
xα
∫ x
0
tα−1L(α−1)
m (−t)L
(α−1)
n−m (t) dt.
(ii) If m ∈ N, α > m− 1, and n ≥ m, then
LII,(α)
m,n (x) = −(α+ n+ 1− 2m)ex
∫ ∞
x
e−tL(−α−1)
m (t)L
(α+1)
n−m (t) dt.
Proof. (i) It follows from [18, equation (17)] that if yn(x) = L
I,(α)
m,n (x), then there exists a poly-
nomial QI
n−m(x) so that
xy′n(x) + αyn(x) = QI
n−m(x)L(α−1)
m (−x). (3.1)
Our first task is to show that QI
n−m(x) = (α+ n)L
(α−1)
n−m (x).
Determinantal Formulas for Exceptional Orthogonal Polynomials 7
Starting from (2.9) and twice using (2.3) we calculate
y′n(x) = − L(α)
m (−x)L
(α)
n−m−1(x) + L
(α+1)
m−1 (−x)L
(α−1)
n−m (x)− L(α−1)
m (−x)L
(α+1)
n−m−2(x)
+ L
(α)
m−1(−x)L
(α)
n−m−1(x)
= L
(α)
n−m−1(x)
(
L
(α)
m−1(−x)− L(α)
m (−x)
)
+ L
(α+1)
m−1 (−x)L
(α−1)
n−m (x)− L(α−1)
m (−x)L
(α+1)
n−m−2(x)
= − L
(α)
n−m−1(x)L
(α−1)
m (−x) + L
(α+1)
m−1 (−x)L
(α−1)
n−m (x)− L(α−1)
m (−x)L
(α+1)
n−m−2(x)
= − L(α−1)
m (−x)
(
L
(α)
n−m−1(x) + L
(α+1)
n−m−2(x)
)
+ L
(α+1)
m−1 (−x)L
(α−1)
n−m (x)
= − L(α−1)
m (−x)L
(α+1)
n−m−1(x) + L
(α+1)
m−1 (−x)L
(α−1)
n−m (x).
Thus
xy′n(x) + αyn(x) = L
(α−1)
n−m (x)
(
xL
(α+1)
m−1 (−x) + αL(α)
m (−x)
)
+ L(α−1)
m (−x)
(
αL
(α)
n−m−1(x)− xL
(α+1)
n−m−1(x)
)
.
We use (2.4) to rewrite the right-hand side of this equation as
L
(α−1)
n−m (x)
(
xL
(α+1)
m−1 (−x) + αL(α)
m (−x)
)
+ (n−m)L(α−1)
m (−x)
(
L
(α)
n−m(x)− L
(α)
n−m−1(x)
)
.
Now we again employ (2.3) to rewrite this as
L
(α−1)
n−m (x)
(
xL
(α+1)
m−1 (−x) + αL(α)
m (−x)
)
+ (n−m)L(α−1)
m (−x)L
(α−1)
n−m (x)
= L
(α−1)
n−m (x)
(
xL
(α+1)
m−1 (−x) + αL(α)
m (−x) + (n−m)L(α−1)
m (−x)
)
.
The proof of the desired formula for QI
n−m will be complete if we can show that
αL(α)
m (t)− tL
(α+1)
m−1 (t) = (m+ α)L(α−1)
m (t),
which follows easily from (2.3) and (2.4).
Consider now the first order linear ODE (3.1) with QI
n−m(x) = (α + n)L
(α−1)
n−m (x). Multiply
both sides by xα−1 and integrate with respect to x to obtain
xαyn(x) = (α+ n)
∫ x
c
tα−1L(α−1)
m (−t)L
(α−1)
n−m (t) dt
for some constant c so that the left-hand side of this expression is equal to 0 when evaluated
at c. Clearly, c = 0 is a valid choice and the desired formula follows.
(ii) Recall [18, equation (41)], which states
d
dx
LII,(α)
m,n (x)− LII,(α)
m,n (x) = (α+ n+ 1− 2m)L(−α−1)
m (x)L
(α+1)
n−m (x) (3.2)
(note that we correct a typo in the constant from [18, equation (41)]). Multiply both sides of
this equation by e−x and integrate to obtain
e−xLII,(α)
m,n (x) = (α+ n+ 1− 2m)
∫ x
c
e−tL(−α−1)
m (t)L
(α+1)
n−m (t) dt (3.3)
for a constant c so that the left-hand side of this expression is equal to 0 when evaluated at c.
Clearly, taking the limit as c → ∞ leads to a valid choice and the desired formula follows. ■
8 B. Simanek
We can use Theorem 3.1 to prove the following determinantal formulas for the exceptional
Laguerre polynomials. Different determinantal formulas for L
I,(α)
m,n (m = 1, 2) involving moments
of an appropriate measure are presented in [20, 21, 26].
Theorem 3.2. For any τ ∈ C, let
{
x
(τ)
j,N
}N
j=1
denote the zeros of L
(τ)
N . Fix m ∈ N and α > 0
and let{
X
(α)
j,m,n
}n
j=1
=
{
−x
(α−1)
j,m
}m
j=1
∪
{
x
(α−1)
j−m,n−m
}n
j=m+1
.
Let M I
n be the (n+ 1)× (n+ 1) matrix whose jth row is[
1,
(
X
(α)
j,m,n
)
,
(
X
(α)
j,m,n
)2
, . . . ,
(
X
(α)
j,m,n
)n]
for j = 1, . . . , n and let the last row of M I
n be[
1
α
,
x
1 + α
,
x2
2 + α
, . . . ,
xn
n+ α
]
.
Then for n ≥ m,
LI,(α)
m,n (x) =
(−1)n−m(α+ n)
m!(n−m)!
∏
1≤i<j≤n
(
X
(α)
j,m,n −X
(α)
i,m,n
) det (M I
n
)
.
Remark 3.3. Notice that the factor
∏
1≤i<j≤n
(
X
(α)
j,m,n −X
(α)
i,m,n
)
is the Vandermonde determi-
nant corresponding to the points
{
X
(α)
j,m,n
}n
j=1
. We will see similar formulas in our later results.
Proof. Write
LI,(α)
m,n (x) =
n∑
j=0
aj,nx
j .
Let M̃ I
n be the (n+ 1)× (n+ 1) matrix whose jth row is[
α,
(
X
(α)
j,m,n
)
(1 + α),
(
X
(α)
j,m,n
)2
(2 + α), . . . ,
(
X
(α)
j,m,n
)n
(n+ α)
]
for j = 1, . . . , n and let the last row of M̃ I
n be
[
1, x, x2, . . . , xn
]
. Equation (3.1) shows that{
X
(α)
j,m,n
}n
j=1
is the zero set of xL
I,(α)
m,n (x)′ + αL
I,(α)
m,n (x). Therefore, if
a⃗ =
a0,n
a1,n
...
an,n
, b⃗I =
0
...
0
L
I,(α)
m,n (x)
,
then M̃ I
na⃗ = b⃗I . It is clear that the determinant of M̃ I
n is a polynomial of degree (at most) n
and the relation M̃ I
na⃗ = b⃗I tells us that if we plug in any zero of L
I,(α)
m,n (x) for x, then a⃗ ∈
Ker
(
M̃ I
n
)
and so det
(
M̃ I
n
)
= 0 at that point. We deduce that det
(
M̃ I
n
)
= C ′L
I,(α)
m,n (x) for some
C ′ ∈ C. It is clear that M I
n is obtained from M̃ I
n by elementary column operations. Therefore,
det
(
M I
n
)
= CL
I,(α)
m,n (x) for some C ∈ C.
Determinantal Formulas for Exceptional Orthogonal Polynomials 9
To determine this constant C, notice that the coefficient of xn in det(M I
n) is
det
1 X
(α)
1,m,n
(
X
(α)
1,m,n
)2 · · ·
(
X
(α)
1,m,n
)n−1
1 X
(α)
2,m,n
(
X
(α)
2,m,n
)2 · · ·
(
X
(α)
2,m,n
)n−1
...
...
...
. . .
...
1 X
(α)
n,m,n
(
X
(α)
n,m,n
)2 · · ·
(
X
(α)
n,m,n
)n−1
· 1
α+n
=
1
α+n
∏
1≤i<j≤n
(
X
(α)
j,m,n −X
(α)
i,m,n
)
.
Observe that the points X
(α)
j,m,n are distinct by part (i) of Theorem 2.1. The desired formula
now follows from (2.1) and (2.9). ■
For our next result, define the polynomials {En(x)}∞n=0 by
En(x) =
n∑
j=0
xj
j!
.
These are the partial sums of the Taylor series for ex around zero and it is also true that
En(x) = (−1)nL
(−n−1)
n (x) (see [22, Section 1.2]).
Theorem 3.4. For any τ ∈ C, let
{
x
(τ)
j,N
}N
j=1
denote the zeros of L
(τ)
N . Fix m ∈ N and α ∈
(m− 1,∞) and let{
Y
(α)
j,m,n
}n
j=1
=
{
x
(−α−1)
j,m
}m
j=1
∪
{
x
(α+1)
j−m,n−m
}n
j=m+1
.
Let M II
n be the (n+ 1)× (n+ 1) matrix whose jth row is[
1, Y
(α)
j,m,n,
(
Y
(α)
j,m,n
)2
, . . . ,
(
Y
(α)
j,m,n
)n]
for j = 1, . . . , n and let the last row of M II
n be
[E0(x), E1(x), 2!E2(x), 3!E3(x), . . . , n!En(x)].
Then for n ≥ m,
LII,(α)
m,n (x) =
(−1)n+1 − (m− α− 1)/(n−m)
m!(n−m− 1)!
∏
1≤i<j≤n
(
Y
(α)
j,m,n − Y
(α)
i,m,n
) det (M II
n
)
.
Proof. Write
LII,(α)
m,n (x) =
n∑
j=0
cj,nx
j .
Let M̃ II
n be the (n+ 1)× (n+ 1) matrix whose jth row is[
1, (Y
(α)
j,m,n − 1),
(
Y
(α)
j,m,n
)
(Y
(α)
j,m,n − 2), . . . ,
(
Y
(α)
j,m,n
)n−1(
Y
(α)
j,m,n − n
)]
for j = 1, . . . , n and let the last row of M̃ II
n be
[
1, x, x2, . . . , xn
]
. Equation (3.2) implies{
Y
(α)
j,m,n
}n
j=1
is the zero set of L
II,(α)
m,n (x)′ − L
II,(α)
m,n (x). Therefore, if
c⃗ =
c0,n
c1,n
...
cn,n
, b⃗II =
0
...
0
L
II,(α)
m,n (x)
,
10 B. Simanek
then M̃ II
n c⃗ = b⃗II . Reasoning as in the proof of Theorem 3.2, we deduce that det
(
M II
n
)
=
CL
II,(α)
m,n (x) for some C ∈ C.
To determine this constant C, notice that the coefficient of xn in det
(
M II
n
)
is
det
1 Y
(α)
1,m,n
(
Y
(α)
1,m,n
)2 · · ·
(
Y
(α)
1,m,n
)n−1
1 Y
(α)
2,m,n
(
Y
(α)
2,m,n
)2 · · ·
(
Y
(α)
2,m,n
)n−1
...
...
...
. . .
...
1 Y
(α)
n,m,n
(
Y
(α)
n,m,n
)2 · · ·
(
Y
(α)
n,m,n
)n−1
=
∏
1≤i<j≤n
(
Y
(α)
j,m,n − Y
(α)
i,m,n
)
.
The desired formula now follows from (2.1), from (2.10), and from Theorem 2.1, which guarantees
that the set
{
Y
(α)
j,m,n
}n
j=1
consists of n distinct points. ■
Our next result is a comparable determinantal formula for the type-III exceptional Laguerre
polynomials. The formula that we will derive is slightly less elegant than those derived in
Theorems 3.2 and 3.4 because we will only have information about zeros of the derivative
of L
III,(α)
m,n and hence will have to account for L
III,(α)
m,n (0) separately.
Theorem 3.5. For any τ ∈ C, let
{
x
(τ)
j,N
}N
j=1
denote the zeros of L
(τ)
N . Fix m ∈ N and α ∈
(−1, 0) and let{
Z
(α)
j,m,n
}n−1
j=1
=
{
−x
(−α−1)
j,m
}m
j=1
∪
{
x
(α+1)
j−m,n−m−1
}n−1
j=m+1
.
Let M III
n be the n× n matrix whose jth row is[
1,
(
Z
(α)
j,m,n
)
,
(
Z
(α)
j,m,n
)2
, . . . ,
(
Z
(α)
j,m,n
)n−1]
for j = 1, . . . , n− 1 and let the last row of M III
n be[
1,
x
2
, . . . ,
xn−1
n
]
.
Then for n > m,
LIII,(α)
m,n (x) =
x(−1)n−m−1n det
(
M III
n
)
m!(n−m− 1)!
∏
1≤i<j≤n−1
(
Z
(α)
j,m,n − Z
(α)
i,m,n
)
+ (m+ 1)
(
n−m+ α
n−m− 1
)(
m− α− 1
m+ 1
)
.
Proof. Write
LIII,(α)
m,n (x) =
n∑
j=0
hj,nx
j .
Let M̃ III
n be the n× n matrix whose jth row is[
1, 2Z
(α)
j,m,n, 3
(
Z
(α)
j,m,n
)2
, . . . , n
(
Z
(α)
j,m,n
)n−1]
for j = 1, . . . , n− 1 and let the last row of M̃ III
n be
[
1, x, x2, . . . , xn−1
]
. Equation (2.12) implies{
Z
(α)
j,m,n
}n−1
j=1
is the zero set of L
III,(α)
m,n (x)′. Therefore, if Pn(x) =
(
L
III,(α)
m,n (x)− h0,n
)
/x and
h⃗ =
h1,n
h2,n
...
hn,n
, b⃗III =
0
...
0
Pn(x)
,
Determinantal Formulas for Exceptional Orthogonal Polynomials 11
then M̃ III
n h⃗ = b⃗III . Reasoning as in the proof of Theorem 3.2, we deduce that det
(
M III
n
)
=
CPn(x) for some C ∈ C.
To determine this constant C, notice that the coefficient of xn−1 in det
(
M III
n
)
is
1
n
det
1 Z
(α)
1,m,n
(
Z
(α)
1,m,n
)2 · · ·
(
Z
(α)
1,m,n
)n−2
1 Z
(α)
2,m,n
(
Z
(α)
2,m,n
)2 · · ·
(
Z
(α)
2,m,n
)n−2
...
...
...
. . .
...
1 Z
(α)
n−1,m,n
(
Z
(α)
n−1,m,n
)2 · · ·
(
Z
(α)
n−1,m,n
)n−2
=
1
n
∏
1≤i<j≤n−1
(
Z
(α)
j,m,n− Z
(α)
i,m,n
)
.
The desired formula now follows from (2.1), from (2.11), from Theorem 2.1 (which guaran-
tees that the set
{
Z
(α)
j,m,n
}n−1
j=1
consists of n− 1 distinct points), and the formula for L
III,(α)
m,n (0)
from (2.12). ■
3.2 Xm Jacobi polynomials
In this section, we will present results for Xm Jacobi polynomials that are analogous to those in
Section 3.1.
Theorem 3.6. Fix m ∈ N and let P
(α,β)
m,n (x) denote the degree n Xm Jacobi polynomial with
β > 0, α > m− 1, and α+ 1−m− β ̸∈ {0, 1, . . . ,m− 1}. Then
P (α,β)
m,n (x) = (−1)m
(β + n)(α+ n− 2m+ 1)
(α+ n−m+ 1)(x+ 1)β
∫ x
−1
(t+ 1)β−1P (−α−1,β−1)
m (t)P
(α+1,β−1)
n−m (t) dt
for all n ≥ m.
Proof. Recall [18, equation (70)] states that
(1 + x)P (α,β)
m,n (x)′ + βP (α,β)
m,n (x)
= (−1)m
(β + n)(α+ n− 2m+ 1)
(α+ n−m+ 1)
P
(α+1,β−1)
n−m (x)P (−α−1,β−1)
m (x). (3.4)
Multiply both sides of (3.4) by (x+ 1)β−1 and integrate to obtain
(x+ 1)βP (α,β)
m,n (x)
= (−1)m
(β + n)(α+ n− 2m+ 1)
(α+ n−m+ 1)
∫ x
c
(t+ 1)β−1P (−α−1,β−1)
m (t)P
(α+1,β−1)
n−m (t) dt
for a constant c so that the left-hand side of this expression is equal to 0 when evaluated at c.
Clearly, c = −1 is a valid choice (because β > 0) and the desired formula follows. ■
For our next result, define the polynomials {Rn(x)}∞n=0 by
Rn(x;β) =
n∑
j=0
β(j)(−x)j
j!
.
These polynomials are the partial sums of the Taylor series for (1 + x)−β around zero and it is
also true that Rn(x, β) = (−1)nP
(−n−1,β)
n (1 + 2x). In our notation, we used the Pochhammer
symbol
a(n) =
{
a(a+ 1)(a+ 2) · · · (a+ n− 1) if n > 0,
1 if n = 0.
12 B. Simanek
Theorem 3.7. Let
{
z
(α,β)
j,N
}N
j=1
denote the zeros of P
(α,β)
N . Choose α, β, m so that the hypotheses
of Theorem 3.6 are satisfied. Let{
W
(α,β)
j,m,n
}n
j=1
=
{
z
(−α−1,β−1)
j,m
}m
j=1
∪
{
z
(α+1,β−1)
j−m,n−m
}n
j=m+1
.
Let Mn be the (n+ 1)× (n+ 1) matrix whose jth row is[
1,W
(α,β)
j,m,n,
(
W
(α,β)
j,m,n
)2
, . . . ,
(
W
(α,β)
j,m,n
)n]
for j = 1, . . . , n and let the last row of Mn be[
1
β(1)
,
−1
β(2)
R1(x),
2!
β(3)
R2(x), . . . ,
(−1)nn!
β(n+1)
Rn(x)
]
.
Then for n ≥ m (with δ = α+ 1),
P (α,β)
m,n (x) =
(−1)m2−n
(
n
m
)
(n+δ−2m)Γ(β−δ+2m)Γ(δ+β+2(n−m))(n+β)
n!(δ + n−m)Γ(β−δ+m)Γ(δ+β+n−m)
∏
i<j
(
W
(α,β)
j,m,n −W
(α,β)
i,m,n
) det(Mn).
Proof. Write
P (α,β)
m,n (x) =
n∑
j=0
dj,nx
j .
Let M̃n be the (n+ 1)× (n+ 1) matrix whose jth row is[
β,
(
1+ (1+ β)W
(α,β)
j,m,n
)
,
(
W
(α,β)
j,m,n
)(
2+ (2+ β)W
(α,β)
j,m,n
)
, . . . ,
(
W
(α,β)
j,m,n
)n−1(
n+ (n+ β)W
(α,β)
j,m,n
)]
for j = 1, . . . , n and let the last row of M̃n be
[
1, x, x2, . . . , xn
]
. Equation (3.4) implies{
W
(α)
j,m,n
}n
j=1
is the zero set of (1 + x)P
(α,β)
m,n (x)′ + βP
(α,β)
m,n (x). Therefore, if
d⃗ =
d0,n
d1,n
...
dn,n
, b⃗ =
0
...
0
P
(α,β)
m,n (x)
,
then M̃nd⃗ = b⃗. Reasoning as in the proof of Theorem 3.2, we deduce that det(Mn) = CP
(α,β)
m,n (x)
for some C ∈ C.
To determine this constant C, notice that the coefficient of xn in det(Mn) is
1
n+ β
det
1 W
(α,β)
1,m,n
(
W
(α,β)
1,m,n
)2 · · ·
(
W
(α,β)
1,m,n
)n−1
1 W
(α,β)
2,m,n
(
W
(α,β)
2,m,n
)2 · · ·
(
W
(α,β)
2,m,n
)n−1
...
...
...
. . .
...
1 W
(α,β)
n,m,n
(
W
(α,β)
n,m,n
)2 · · ·
(
W
(α,β)
n,m,n
)n−1
=
1
n+β
∏
1≤i<j≤n
(
W
(α,β)
j,m,n −W
(α,β)
i,m,n
)
.
The desired formula now follows from (2.5), from (2.13), and from Theorem 2.2, which guarantees
us that the set
{
W
(α,β)
j,m,n
}n
j=1
consists of n distinct points. ■
Determinantal Formulas for Exceptional Orthogonal Polynomials 13
3.3 X2 Hermite polynomials
The starting point for our work with these polynomials comes from [14, Proposition 5.4], which
states that if a polynomial p is in the span of
{
H
(λ)
|λ|,n
}
n∈Nλ
, then
2H ′
λ(xp− p′) +H ′′
λp
is divisible by Hλ (note that the complete statement of [14, Proposition 5.4] was corrected in
[7, 15]). Since each polynomial in the sequence
{
H
(λ)
|λ|,n
}
n∈Nλ
is obviously in the span of that
sequence, it is interesting to find the polynomial Qn,λ such that
2H ′
λ(x)
(
xH
(λ)
|λ|,n(x)−H
(λ)
|λ|,n(x)
′)+H ′′
λ(x)H
(λ)
|λ|,n(x) = Qn,λ(x)Hλ(x). (3.5)
The search for such a polynomial leads us to the following result.
Theorem 3.8. Let λ = {1, 1} and let H
({1,1})
2,n denote the corresponding degree n exceptional
Hermite polynomial with n ≥ 3. Then
H
({1,1})
2,n (x) = 8n(n− 1)(n− 2)
∫ x
0
H{1,1}(t)Hn−3(t) dt+ 16(n− 1)(n− 2)Hn−2(0)
=
∫ x
0
H{1,1}(t)H
′′′
n (t) dt+ 8(n− 2)H ′
n−1(0).
Proof. It suffices to show that
d
dx
H
({1,1})
2,n (x) = 8n(n− 1)(n− 2)H{1,1}(x)Hn−3(x),
and to calculate H
({1,1})
2,n (0). This follows easily from the formulas in [29, Section 6.4], namely
H{1,1}(x) = 4
(
2x2 + 1
)
and
H
({1,1})
2,n (x) = det
2x 4x2 − 2 Hn(x)
2 8x H ′
n(x)
0 8 H ′′
n(x)
for n ∈ N{1,1}. As mentioned there, the formulas (2.7) and (2.8) imply
H
({1,1})
2,n (x) = 16(n− 1)
(
−2xHn−1(x) +
(
n
(
2x2 + 1
)
− 2
)
Hn−2(x)
)
(3.6)
and from here, the first equality in the theorem is an elementary calculation using (2.7) and (2.8).
The second formula follows from (2.7). ■
Numerical experiments have confirmed that for partitions other than {1, 1}, it is not neces-
sarily true that Hλ divides H
(λ)
|λ|,n(x)
′. One can use the result of [14, Proposition 5.4] mentioned
above to find a first order differential relation for H
(λ)
|λ|,n(x), but the coefficients will be left in
terms ofHλ, which is difficult to calculate explicitly and hence does not lead to a nice integral for-
mula as in Theorem 3.8. Even still, it would be interesting to characterize the polynomials Qn,λ
from (3.5).
Following the method of the previous sections, we can use Theorem 3.8 to prove a determi-
nantal formula for H
({1,1})
2,n .
14 B. Simanek
Theorem 3.9. Let {yj,N}Nj=1 denote the zeros of the classical Hermite polynomial HN and let
{Uj,n}n−1
j=1 =
{
i√
2
,
−i√
2
}
∪ {yj−2,n−3}n−1
j=3 .
Let M
{1,1}
n be the n× n matrix whose jth row is[
1, (Uj,n), (Uj,n)
2, . . . , (Uj,n)
n−1
]
for j = 1, . . . , n− 1 and let the last row of M
{1,1}
n be[
1,
x
2
, . . . ,
xn−1
n
]
.
Then for n ≥ 3
H
({1,1})
2,n (x) = x
2n+3n(n− 1)(n− 2)∏
1≤i<j≤n−1(Uj,n − Ui,n)
det
(
M{1,1}
n
)
+ 16(n− 1)(n− 2)Hn−2(0).
Remark 3.10. Notice the similarity to Theorem 3.5 in that the constant term of the polynomial
needs to be added in separately because the determinant alone only gives a polynomial of
degree n− 1.
Proof. Write
H
({1,1})
2,n (x) =
n∑
j=0
tj,nx
j .
Let M̃
{1,1}
n be the n× n matrix whose jth row is[
1, 2Uj,n, 3(Uj,n)
2, . . . , n(Uj,n)
n−1
]
for j = 1, . . . , n − 1 and let the last row of M̃
{1,1}
n be
[
1, x, x2, . . . , xn−1
]
. Theorem 3.8 implies
{Uj,n}n−1
j=1 is the zero set of H
({1,1})
2,n (x)′. Therefore, if Qn(x) =
(
H
({1,1})
2,n (x)− t0,n
)
/x and
t⃗ =
t1,n
t2,n
...
tn,n
, b⃗{1,1} =
0
...
0
Qn(x)
,
then M̃
{1,1}
n t⃗ = b⃗{1,1}. Reasoning as in the proof of Theorem 3.2, we deduce that det
(
M
{1,1}
n
)
=
CQn(x) for some C ∈ C.
To determine this constant C, notice that the coefficient of xn−1 in det
(
M
{1,1}
n
)
is
1
n
det
1 U1,n (U1,n)
2 · · · (U1,n)
n−2
1 U2,n (U2,n)
2 · · · (U2,n)
n−2
...
...
...
. . .
...
1 Un−1,n (Un−1,n)
2 · · · (Un−1,n)
n−2
=
1
n
∏
1≤i<j≤n−1
(Uj,n − Ui,n).
The desired formula now follows from (2.6) and (3.6). ■
Determinantal Formulas for Exceptional Orthogonal Polynomials 15
Acknowledgements
It is a pleasure to thank Rob Milson for helpful discussion about the content of this work.
The author gratefully acknowledges support from the Simons Foundation through collaboration
grant 707882. The author also thanks the anonymous referees for their useful feedback.
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1 Introduction
2 Background and formulas
2.1 Classical orthogonal polynomials
2.1.1 Classical Laguerre polynomials
2.1.2 Classical Jacobi polynomials
2.1.3 Classical and generalized Hermite polynomials
2.2 Exceptional orthogonal polynomials
2.2.1 Exceptional X_m-Laguerre polynomials
2.2.2 Exceptional X_m-Jacobi polynomials
2.2.3 Exceptional X_m-Hermite polynomials
3 New formulas
3.1 X_m Laguerre polynomials
3.2 X_m Jacobi polynomials
3.3 X_2 Hermite polynomials
References
|
| id | nasplib_isofts_kiev_ua-123456789-211621 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T02:36:27Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Simanek, Brian 2026-01-07T13:39:36Z 2022 Determinantal Formulas for Exceptional Orthogonal Polynomials. Brian Simanek. SIGMA 18 (2022), 047, 16 pages 1815-0659 2020 Mathematics Subject Classification: 42C05; 33C47 arXiv:2202.01278 https://nasplib.isofts.kiev.ua/handle/123456789/211621 https://doi.org/10.3842/SIGMA.2022.047 We present determinantal formulas for families of exceptional ₘ-Laguerre and exceptional ₘ-Jacobi polynomials and also for exceptional ₂-Hermite polynomials. The formulas resemble Vandermonde determinants and use the zeros of the classical orthogonal polynomials. It is a pleasure to thank Rob Milson for the helpful discussion about the content of this work. The author gratefully acknowledges support from the Simons Foundation through a collaboration grant 707882. The author also thanks the anonymous referees for their useful feedback. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Determinantal Formulas for Exceptional Orthogonal Polynomials Article published earlier |
| spellingShingle | Determinantal Formulas for Exceptional Orthogonal Polynomials Simanek, Brian |
| title | Determinantal Formulas for Exceptional Orthogonal Polynomials |
| title_full | Determinantal Formulas for Exceptional Orthogonal Polynomials |
| title_fullStr | Determinantal Formulas for Exceptional Orthogonal Polynomials |
| title_full_unstemmed | Determinantal Formulas for Exceptional Orthogonal Polynomials |
| title_short | Determinantal Formulas for Exceptional Orthogonal Polynomials |
| title_sort | determinantal formulas for exceptional orthogonal polynomials |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211621 |
| work_keys_str_mv | AT simanekbrian determinantalformulasforexceptionalorthogonalpolynomials |