# Exceptional Gegenbauer polynomials via isospectral deformation

María Ángeles García-Ferrero  BCAM - Basque Center for Applied Mathematics,
Alameda de Mazarredo 14, 48009 Bilbao, Spain<EMAIL_ADDRESS>,  David
Gómez-Ullate  Departamento de Ingeniería Informática, Escuela Superior de
Ingeniería, Universidad de Cádiz, 11519 Puerto Real, Spain.  Departamento de
Física Teórica, Universidad Complutense de Madrid, 28040 Madrid, Spain.
<EMAIL_ADDRESS>,  Robert Milson  Department of Mathematics and
Statistics, Dalhousie University, Halifax, NS, B3H 3J5, Canada.
<EMAIL_ADDRESS>and  James Munday  Department of Mathematics and Statistics,
Dalhousie University, Halifax, NS, B3H 3J5, Canada<EMAIL_ADDRESS>
(Date: Updated September 29, 2021)

###### Abstract.

We show a method to construct isospectral deformations of classical orthogonal
polynomials. The construction is based on confluent Darboux transformations,
and it allows to construct Sturm-Liouville problems with polynomial
eigenfunctions that have an arbitrary number of continuous parameters. We
propose to call these new orthogonal polynomial systems _exceptional
polynomials of the second kind_. We illustrate this construction by describing
the class of exceptional Gegenbauer polynomials of the second kind.

###### Contents

  1. 1 Introduction
  2. 2 Darboux transformations and factorization chains
  3. 3 Confluent Darboux transformations
  4. 4 Exceptional Gegenbauer operators and polynomials
  5. 5 Exceptional Gegenbauer polynomials of the second kind
  6. 6 Examples

##  1\. Introduction

Classical orthogonal polynomials (OPs) have been traditionally characterized
as the only orthogonal polynomial bases of an $\rm L^{2}$ space that are also
eigenfunctions of a Sturm-Liouville problem. Since the mid-nineteenth century
until today, classical OPs appear in ubiquitous applications in mathematical
physics, numerical analysis, approximation theory or statistics, among other
fields.

As it is well known, classical OPs can be classified into three main families,
depending on whether they are defined on the whole real line (Hermite), the
half-line (Laguerre) or a compact interval (Jacobi). These polynomial families
are characterized by two free real parameters in the case of Jacobi, one for
Laguerre and none for Hermite.

This article tackles the following question: _is it possible to construct an
orthogonal basis of an $\rm L^{2}$ space, which is also formed by polynomial
eigenfunctions of a Sturm-Liouville problem, but contains a higher (possibly
arbitrary) number of free real parameters?_

The flexibility of deforming classical OPs to contain so many free real
parameters and yet mantain many of their defining properties would open the
way to many potential applications in all of the fields where classical OPs
appear naturally.

In this paper we show how to construct Sturm-Liouville problems with
polynomial eigenfunctions that contain an arbitrary number of real parameters,
thus providing a positive answer to the previous question. More specifically,
let $T$ be a _classical differential operator_ , i.e. a second-order
differential operator whose eigenfunctions are classical OPs,

$T(z,D_{z}):=p(z)D_{z}^{2}+q(z)D_{z}+r(z),$

and let $\\{P_{i}\\}_{i=0}^{\infty}$ be a set of polynomial eigenfunctions of
$T$

$TP_{i}=\lambda_{i}P_{i},\quad i=0,1,2,\dots$

We will show how to construct a new operator $\hat{T}$ and polynomials
$\hat{P}_{i}$ such that

$\hat{T}(z,D_{z};t_{m_{1}},\dots,t_{m_{n}}):=p(z)D_{z}^{2}+\hat{q}(z;t_{m_{1}},\dots,t_{m_{n}})D_{z}+\hat{r}(z;t_{m_{1}},\dots,t_{m_{n}})$

where $t_{m_{1}},\dots,t_{m_{n}}$ are $n$ real parameters, $\hat{q}$ and
$\hat{r}$ are rational functions of $z$, and

$\hat{T}\hat{P}_{i}=\lambda_{i}\hat{P}_{i},\quad i\in\\{0,1,2,\dots\\}.$

The transformed eigenvalue problem has the same spectrum
$\\{\lambda_{i}\\}_{i=0}^{\infty}$ as the original one, and the leading
coefficient $p(z)$ does not change in the transformation. We speak thus of a
_isospectral deformation_ because we will also see that

$\hat{T}(z,D_{z};0,\dots,0)=T(z,D_{z}),\qquad\hat{P}_{i}(z;0,\dots,0)=P_{i}(z).$

So much extra freedom and flexibility comes at a cost. Although the set of new
polynomials is still $\mathrm{L}^{2}$-complete, there is a finite number of
exceptional degrees for which no polynomial of that degree exists in the
basis. In general we will have that

$i=\deg P_{i}\neq\deg\hat{P}_{i},\quad i\in\\{0,1,2,\dots\\}$

The new polynomial families introduced in this paper fit thus into the
definition of _exceptional orthogonal polynomials_ (XOPs) originally
introduced in [20]. In the past ten years, the mathematical study of XOPs has
been concerned with their classification [12, 18], the properties of their
zeroes [22, 25, 30], and their recurrence relations [9, 21, 32, 33]. The
spectral-theoretic aspects of exceptional operators (the second-order
differential operators whose eigenfunctions are the XOPs) have been studied in
[28, 29, 16]. It is known that every exceptional operator can be related to a
classical Bochner operator by a finite number of state-adding, state-deleting,
or isospectral Darboux transformations [12, Theorem 1.2].

In the context of mathematical physics, XOPs appear as exact solutions to
Dirac’s equation [36] and as bound states of exactly-solvable rational
extensions [15, 34, 35]. Additionally, they are connected to finite-gap
potentials [26] and super-integrable systems [31, 35].

The contribution of this paper is to present a new, different construction of
exceptional orthogonal polynomials, that we propose to call (rather
unimaginatively) exceptional orthogonal polynomials of the second kind. There
are a number of important differences between the class of known XOPs (of the
first kind) and the new XOPs of the second kind, including:

  1. (a)

XOPs of the first kind have the same number of real parameters as their
classical counterparts, while XOPs of the second kind have as many real
parameters as desired.

  2. (b)

XOPs of the first kind are not a continuous deformation of a classical
polynomial family, while XOPs of the second kind become classical when we set
all deformation parameters to zero.

  3. (c)

XOPs of the first kind have a different spectrum with respect to their
classical counterparts while XOPs of the second kind have the same spectrum.

  4. (d)

The degree sequence for XOPs of the first kind is an increasing sequence,
which is no longer true for XOPs of the second kind.

As an illustration of this new construction, we show how to define an
isospectral deformation of the classical Gegenbauer operator

(1) $T^{(\alpha)}:=(1-z^{2})D_{z}^{2}-(2\alpha+1)zD_{z}$

through the application of a finite number of confluent Darboux
transformations (CDTs), also referred to as the “double commutator” method
[14].

Exceptional Jacobi polynomials of the first kind are indexed by discrete
parameters [5, 10] and, as a result, cannot be continuously deformed into
their classical counterparts. By contrast, every CDT introduces a new
deformation parameter. Therefore, by performing a chain of $n$ CDTs on the
classical Gegenbauer operator (1) at distinct eigenvalues, we will arrive at
an exceptional operator that depends on $n$ discrete parameters and $n$ real
parameters. The eigenpolynomials of the resulting exceptional operator are
exceptional Gegenbauer polynomials of the second kind, which also depend on
$n$ real parameters, and can be continuously deformed to the classical
Gegenbauer polynomials by letting the parameters tend to zero.

The new construction of exceptional polynomials and weights described in this
paper can also be understood from the point of view of the theory of inverse
scattering [1, 6, 37], and it is conceptually related to the construction of
KdV multi-solitons. While KdV solitons are obtained by applying a state-adding
deformation on the zero potential, the exceptional operators in this paper are
related to isospectral deformations of particular instances of the
Darboux–Pöschl–Teller (DBT) potential [23]. In spectral-theoretic terms, the
consequence of our construction is the continuous modification of the norming
constants of a finite number of the bound states. However, since our focus is
on orthogonal polynomials, rather than quantum mechanics or evolution
equations, our approach is formulated in the gauge and coordinates of the
classical differential operator rather than working with Schrödinger
operators. The resulting procedure can be easily implemented using a computer
algebra system.

For the remainder of the paper, we develop the theory of confluent Darboux
transformations in the algebraic gauge, and apply it in a construction of new
exceptional Gegenbauer polynomials of the second kind. The paper is organized
as follows: in Section 2 we describe the formal theory of rational multi-step
Darboux-Crum transformations in the algebraic gauge. In Section 3 we describe
confluent Darboux transformations as a 2-step Darboux transformation whose
seed functions are an eigenfunction and a generalized eigenfunction at the
same eigenvalue. In Section 4, we describe the class of exceptional Gegenbauer
operators and their factorizations and we provide a recursive construction for
the operators and eigenfunctions connected by CDTs. In Section 5 we provide
matrix formulas for exceptional Gegenbauer polynomials of second type, we
prove the equivalence of the matrix and recursive definitions and thereby
establish the proofs of our main results concerning the Sturm–Liouville
properties and $\mathrm{L}^{2}$-completeness of the exceptional Gegenbauer
polynomials. Finally, in Section 6, we provide explicit examples of families
of exceptional Gegenbauer polynomials of the second kind, for both one and two
deformation parameters.

Although this is the first paper to provide a detailed description of the
construction of exceptional polynomials of the second kind via confluent
Darboux transformations, we would like to mention that exceptional polynomials
obtained by confluent Darboux transformations were investigated earlier in the
1-step case (with one deformation parameter) by Grandati and Quesne [24]. The
family of exceptional Legendre polynomials of the second kind was recently
introduced by some of the present authors [13], showing for the first time
that deformations of classical polynomials with an arbitrary number of real
parameters exist. Shortly after that, Durán has shown that there is an
alternative way to construct exceptional polynomials of the second kind, by
first perturbing the measure of the discrete Hahn polynomials, dualizing and
taking a suitable limit[11]. The extra continuous parameters enter in this
construction in the polynomial perturbation of the classical discrete measure.
Summarizing, there are three rather different constructions leading to the
same mathematical objects:

  1. (i)

one based in iterating the action of differential operators, leading to
Wronskian determinants whose seed functions are generalized eigenfunctions,

  2. (ii)

one based in matrix and integral formulas, coming from the inverse scattering
method,

  3. (iii)

one based in Durán’s bispectral construction, coming from a perturbation of a
classical discrete measure.

In this paper we show the equivalence of the first two approaches. Although
there is a strong evidence that the last construction leads to the same
objects, establishing the equivalence of all three methods remains an open
question, which can possibly trigger further research.

### Notation

Throughout the paper we use $\mathbb{N}=\\{1,2,\ldots\\}$ to denote the set of
natural numbers and $\mathbb{N}_{0}=\\{0,1,\dots\\}$ to denote the set of non-
negative integers. We use “half-integer” to refer to an odd integer divided by
$2$. The set of positive half-integers will be denoted by
$\mathbb{N}_{0}+\frac{1}{2}$.

We let $D_{z}$ denote the derivative with respect to $z$. For the sake of
notational convenience, we will often drop the explicit dependence on the
indeterminate $z$ and write $\phi=\phi(z),\;\phi^{\prime}=D_{z}\phi=\phi_{z}$,
and $D=D_{z}$.

We call a differential expression of the form
$\sum_{k=0}^{n}p_{k}(z)D_{z}^{k},$ where $p_{0}(z),\ldots,p_{n}(z)$ are
rational functions and $p_{n}\neq 0$, an _$n^{\text{th}}$ -order rational
operator_. We will call a function $\phi(z)$ quasi-rational if its log-
derivative $w(z)=\phi^{\prime}(z)/\phi(z)$ is rational. We denote by
$\operatorname{\operatorname{Wr}}[y_{1},\dots,y_{k}]$ the Wronskian
determinant of the functions $y_{1},\dots,y_{k}$.

We will denote matrices by calligraphic symbols, such as ${\mathcal{R}}$,
whereas 1-dimensional tuples will be given bold symbols such as
${\boldsymbol{m}}$, $\boldsymbol{t}$, or ${\boldsymbol{Q}}$. To access the
components of a vector or tensor we will employ square brackets, i.e.
$[{\mathcal{R}}]_{k\ell}$ denotes the $(k,\ell)$ entry of ${\mathcal{R}}$.

We will denote an $n$–tuple of integers by
${\boldsymbol{m}}=(m_{1},\dots,m_{n})\in\mathbb{N}_{0}^{n}$, and associate to
it the $n$–tuple of real parameters
$\boldsymbol{t}_{{\boldsymbol{m}}}=(t_{m_{1}},\dots,t_{m_{n}})\in\mathbb{R}^{n}$.
We will separate objects of different natures, such as real parameters and
tuples, by semicolons. The concatenation of tuples will be shown using commas,
e.g. if $i_{1},\dots,i_{k}\in\mathbb{N}_{0}$, then
$({\boldsymbol{m}},i_{1},\dots,i_{k})$ denotes the $(n+k)$–tuple
$(m_{1},\dots,m_{n},i_{1},\dots,i_{k})$. We will frequently omit parentheses
when denoting $1$-tuples, opting to write $m_{1}$ instead of $(m_{1})$.
Occasionally, we will omit the dependence on the parameter $\alpha$, so as not
to conflict with other superscript notations.

##  2\. Darboux transformations and factorization chains

The formal theory of Darboux transformations for Schödinger operators in
mathematical physics (also known as supersymmetric quantum mechanics) has been
developed in numerous works, mostly with the aim of generating new solvable
problems from known ones, [7, 6, 19], but also in the construction of
solutions to Painlevé type equations, [3, 17]. It was recently shown that
every exceptional polynomial family must be related to a classical family by a
sequence of Darboux transformations, [12]. We first describe here Darboux
transformations with seed functions that have no repeated eigenvalues. This is
the class of transformations that leads to exceptional orthogonal polynomials
of the first kind, when applied on the classical polynomial families, [5, 9,
10, 15, 18].

In this Section we revise some of these well known results, albeit with a
little twist: we describe Darboux transformations for the class of general
second order differential operators, which include Schrödinger operators as a
particular case. For the purpose of this paper, we focus on second order
differential operators with rational coefficients that have an infinite number
of polynomial eigenfunctions, and we restrict to rational Darboux
transformations of these operators that preserve this property by
construction. However, the results derived in this section can be trivially
extended to general second order operators. All of this section is written at
a purely formal level, so in an abuse of notation we speak of differential
operators without specifying their domain, or we speak of eigenfunctions as
solutions of an eigenvalue problem, without defining a proper spectral
theoretic setting.

######  Definition 2.1.

For $n\in\mathbb{N}_{0}$, let $T_{0},T_{1},\ldots,T_{n}$ be second-order
rational operators. We say that $T_{0}\to T_{1}\to\cdots\to T_{n}$ is an _$n$
-step rational Darboux transformation_ if there exist first-order rational
operators $A_{1},A_{2},\ldots,A_{n}$ such that

(2) $A_{k}T_{k-1}=T_{k}A_{k},\quad k=1,\ldots,n.$

The next Proposition states that if $A_{k}$ is a first order intertwiner
operator between $T_{k-1}$ and $T_{k}$ as in (2), then its kernel must be
spanned by a formal eigenfunction of $T_{k-1}$.

######  Proposition 2.2.

For $n\in\mathbb{N}_{0}$, let $T_{0}\to T_{1}\to\dots\to T_{n}$ be an $n$-step
rational Darboux transformation, and $A_{1},\ldots,A_{n}$ be the first-order
rational operators satisfying the intertwining relation (2). Let $b_{k}(z)$
and $w_{k}(z)$ be rational functions such that

(3) $A_{k}=b_{k}\left(D-w_{k}\right),$

and define (up to a constant factor) the quasi-rational functions

(4) $\psi_{k}(z):=\exp\left(\int^{z}w_{k}(u)du\right).$

We then have that

(5) $T_{k-1}\psi_{k}=\lambda_{k}\psi_{k},\qquad k=1,\ldots,n,$

where $\lambda_{1},\lambda_{2},\ldots,\lambda_{n}$ are constants.

###### Proof.

By (3) and (4)we have that $A_{k}\psi_{k}=0$, so (2) implies that
$A_{k}T_{k-1}\psi_{k}=0$. Since $A_{k}$ is a first-order operator, its kernel
is 1-dimensional and it is therefore spanned by $\psi_{k}$. This implies that
there exists a constant $\lambda_{k}$ such that (5) holds. ∎

######  Remark 2.3.

Assume that $T_{k}$ has the form

(6) $T_{k}=pD^{2}+q_{k}D+r_{k},\quad k=0,1,\ldots,n.$

Observe that its coefficients $p,q_{k}$ and $r_{k}$ are related to the
rational functions $w_{1},\ldots,w_{n}$ defined in (3) by the following
Ricatti-type equations

(7) $p(w_{k}^{\prime}+w_{k}^{2})+q_{k-1}w_{k}+r_{k-1}=\lambda_{k},\quad
k=1,\ldots,n.$

For a given $n$-step Darboux transformation, the corresponding
$\psi_{1},\ldots,\psi_{n}$ are unique, up to a choice of multiplicative
constant. In light of this remark and future ones, we introduce the following
defintions:

######  Definition 2.4.

We call the quasi-rational functions $\psi_{1},\ldots,\psi_{n}$ _factorization
eigenfunctions_ , the set of numbers $\lambda_{1},\ldots,\lambda_{n}$
_factorization eigenvalues_ , and the rational functions $b_{1},\ldots,b_{n}$
_factorization gauges_.

A chain of Darboux transformations is more often defined as a factorization of
each second-order operator $T_{k}$ for $k=0,\ldots,n$, followed by a
permutation of the two factors to yield the next operator $T_{k+1}$. We make
precise the notion of a factorization chain in the next definition and
establish later that both approaches (intertwining of operators and
factorization) coincide.

######  Definition 2.5.

Let $T_{0},T_{1},\ldots,T_{n}$ be second-order rational operators. We say that
$T_{0}\to T_{1}\to\cdots\to T_{n}$ is a _factorization chain_ if there exist
first-order rational operators $A_{k},B_{k},\;k=1,\ldots,n$ and constants
$\lambda_{1},\ldots,\lambda_{n}$ such that

(8) $T_{k-1}=B_{k}A_{k}+\lambda_{k},\quad T_{k}=A_{k}B_{k}+\lambda_{k},\qquad
k=1,\ldots,n,$

The intertwining relations given in (2) then follow as a consequence. We now
show that these two formulations of Darboux transformations are equivalent.

######  Proposition 2.6.

A multi-step rational Darboux transformation is necessarily a factorization
chain, and viceversa.

###### Proof.

Suppose that (8) holds. Then, the intertwining relations (2) follow by the
associativity of operator composition.

Conversely, suppose that (2) holds, let $A_{k}$ be as in (3), and set

(9) $\displaystyle B_{k}:=\hat{b}_{k}(D-\hat{w}_{k}),$

where

(10)
$\displaystyle\hat{w}_{k}:=-w_{k}-\frac{q_{k-1}}{p}+\beta_{k},\quad\beta_{k}:=\frac{b_{k}^{\prime}}{b_{k}},\quad\hat{b}_{k}:=\frac{p}{b_{k}}.$

Let $\lambda_{1},\ldots,\lambda_{n}$ denote the eigenvalues as per (5). By (7)
and a direct calculation, we have

(11) $\displaystyle B_{k}A_{k}$
$\displaystyle=\hat{b}_{k}(D-\hat{w}_{k})b_{k}(D-w_{k})$
$\displaystyle=p(D-\hat{w}_{k}+\beta_{k})(D-w_{k})$
$\displaystyle=p(D+w_{k})(D-w_{k})+q_{k-1}(D-w_{k})$
$\displaystyle=pD^{2}+q_{k-1}D-p(w_{k}^{\prime}+w_{k}^{2})-q_{k-1}w_{k}$
$\displaystyle=T_{k-1}-\lambda_{k}.$

Hence, it follows that

$\displaystyle(T_{k}-A_{k}B_{k}-\lambda_{k})A_{k}=A_{k}(T_{k-1}-B_{k}A_{k}-\lambda_{k})=0.$

The ring of differential operators does not have zero divisors, so relations
(8) follow immediately. ∎

######  Remark 2.7.

As a direct consequence we may observe that Darboux transformations are
invertible. Indeed, by (8), we have that

$\displaystyle B_{k}T_{k}=T_{k-1}B_{k},\quad k=1,\ldots,n,$

where $B_{1},\ldots,B_{n}$ are the rational operators defined by (9) and (10).
Thus, the chain $T_{n}\to T_{n-1}\to\cdots\to T_{0}$ also satisfies the
definition of an $n$-step Darboux transformation.

###  2.1. Seed eigenfunctions and generalized Crum formula

So far we have defined $n$-step Darboux transformations referring to the
intertwining or factorization of operators $T_{0},T_{1},\dots,T_{n}$ at each
step of the chain, which requires an eigenfunction $\psi_{k}$ for each
operator $T_{k-1}$ in the chain. In this section we show how to define an
$n$-step Darboux transformation using only eigenfunctions of the first
operator $T_{0}$, that we will call _seed functions_. In the case of Darboux
chains for Schrödinger operators, this construction leads to the well known
Darboux-Crum Wronskian formula, [8]. Theorem 2.1 in this section can thus be
seen as a generalization of Crum’s formula.

Let $T_{0}$ be a second-order rational operator and let
$\phi_{1},\phi_{2},\ldots,\phi_{n}$ be quasi-rational eigenfunctions of
$T_{0}$. Explicitly, we have

$\displaystyle T_{0}\phi_{k}=\lambda_{k}\phi_{k},\quad k=1,\ldots,n,$

where $\lambda_{1},\lambda_{2},\ldots,\lambda_{n}$ are constants. We refer to
$\phi_{1},\ldots,\phi_{n}$ as _seed eigenfunctions_ , because, as we show
below, a Darboux transformation is determined by a choice of seed
eigenfunctions and factorization gauges.

Going forward, for a set of indices
$\\{i_{1},\ldots,i_{k}\\}\subseteq\\{1,\ldots,n\\}$, we write

(12)
$\phi_{(i_{1},i_{2},\ldots,i_{k})}:=\operatorname{\operatorname{Wr}}[\phi_{i_{1}},\phi_{i_{2}},\ldots,\phi_{i_{k}}].$

We now arrive at the key result of this section: explicit formulas for the
coefficients of an operator obtained by a multi-step Darboux transformation.

######  Theorem 2.1.

Let $T_{0}$ be a second-order rational operator as in (6),
$\\{\phi_{1},\ldots,\phi_{n}\\}$ be quasi-rational eigenfunctions of $T_{0}$
with distinct eigenvalues $\lambda_{1},\ldots,\lambda_{n}$ and let
$\\{b_{1},\ldots,b_{n}\\}$ be a set of non-zero rational functions. Define the
operators

(13) $T_{k}:=pD^{2}+q_{k}D+r_{k},\qquad A_{k}:=b_{k}(D-w_{k}).$

where

(14) $\sigma_{k}:=\sum_{j=1}^{k}(\log
b_{j})^{\prime},\quad\upsilon_{k}:=\frac{\phi_{(1,2,\ldots,k)}^{\prime}}{\phi_{(1,2,\ldots,k)}},\qquad
k=1,\ldots,n;$

and

(15) $\displaystyle\begin{split}&q_{k}:=q_{0}+kp^{\prime}-2p\sigma_{k},\\\
&r_{k}:=r_{0}+kq_{0}^{\prime}+\frac{1}{2}k(k-1)p^{\prime\prime}+\upsilon_{k}p^{\prime}-\sigma_{k}(q_{0}+kp^{\prime})+(\sigma_{k}^{2}-\sigma_{k}^{\prime}+2\upsilon_{k}^{\prime})p,\end{split}$
(16) $\displaystyle\begin{split}&w_{1}:=\upsilon_{1},\quad
w_{k}:=\sigma_{k-1}+\upsilon_{k}-\upsilon_{k-1},\quad k=2,\dots,n.\end{split}$

Then, the sequence of operators $T_{0}\to T_{1}\to\cdots\to T_{n}$ is an
$n$-step Darboux transformation in the sense of Definition 2.1, i.e. (2) holds
with the operators $A_{k}$ defined above.

Since we assume that the eigenvalues $\lambda_{1},\ldots,\lambda_{n}$ are
distinct, $\phi_{1},\ldots,\phi_{n}$ are linearly independent. Hence, the
Wronskians in the denominator of (14) are non-zero, and the functions
$\upsilon_{k}$ are well-defined.

######  Remark 2.8.

The well-known Crum formula for an $n$-step Darboux transformation of a
Schrödinger operator

$T_{n}=T_{0}-2(\log\phi_{(1,2,\ldots,n)})^{\prime\prime}$

is a special case of Theorem 2.1 that corresponds to starting from a
Schrödinger operator ($p=-1$ and $q_{0}=0$) and choosing the gauge
$b_{1}=\ldots=b_{n}=1$. Thus, this result can be regarded as an extension of
the Darboux-Crum formula from Schrödinger operators to general second order
operators. Although for the purpose of the paper we restrict to second order
operators with rational coefficients and polynomial eigenfunctions, the
generalized Crum formula in Theorem 2.1 is valid on a general setting.

The rest of this section is devoted to the proof of Theorem 2.1. The strategy
is to first establish the result for the particular gauge
$b_{1}=\cdots=b_{n}=1$ and then show the transformation rules under a
different gauge. Most of the proofs proceed by induction and make use of
differential algebra and properties of Wronskian determinants. We will first
need to state and prove a number of auxiliary lemmas, where operators with
tilde denote the restriction of the same objects in Theorem 2.1 to the special
case $b_{1}=\cdots=b_{n}=1$.

######  Lemma 2.9.

Under the same setting as in Theorem 2.1, let $\tilde{A}_{k}=D-\tilde{w}_{k}$
for $k\in\\{1,\dots,n\\}$ with $\tilde{w}_{k}=\upsilon_{k}-\upsilon_{k-1}$.
Then,

(17)
$\displaystyle(\tilde{A}_{k-1}\tilde{A}_{k-2}\cdots\tilde{A}_{1})\phi_{j}=\frac{\phi_{(1,2,\ldots,k-1,j)}}{\phi_{(1,2,\ldots,k-1)}},\quad
2\leq k\leq j\leq n.$

###### Proof.

For convenience, set

(18)
$\displaystyle\varphi_{k{\color[rgb]{0,0,1},}j}:=(\tilde{A}_{k-1}\tilde{A}_{k-2}\cdots\tilde{A}_{1})\phi_{j}.$

We argue by induction. The case of $k=2$ follows directly from the definition.
Indeed,

$\displaystyle\varphi_{2,j}=\tilde{A}_{1}\phi_{j}=\frac{1}{\phi_{1}}\operatorname{\operatorname{Wr}}[\phi_{1},\phi_{j}]=\frac{\phi_{1,j}}{\phi_{1}}.$

Now, we suppose that (17) holds for a particular $k<n$. By the inductive
hypothesis, in particular we have

$\varphi_{k,k}=\frac{\phi_{(1,2,\ldots,k)}}{\phi_{(1,2,\ldots,k-1)}},\quad\tilde{\omega}_{k}=\frac{\varphi_{k,k}^{\prime}}{\varphi_{k,k}}\quad\mbox{
and }\quad\tilde{A}_{k}=D-\frac{\varphi_{k,k}^{\prime}}{\varphi_{k,k}}.$

Hence,

$\displaystyle\varphi_{k+1,j}$
$\displaystyle=(\tilde{A}_{k}\cdots\tilde{A}_{1})\phi_{j}=\tilde{A}_{k}\varphi_{k,j}$
$\displaystyle=\frac{1}{\varphi_{k,k}}\operatorname{\operatorname{Wr}}\left[\varphi_{k,k},\varphi_{k,j}\right]=\frac{1}{\varphi_{k,k}}\operatorname{\operatorname{Wr}}\left[\frac{\phi_{(1,2,\ldots,k-1,k)}}{\phi_{(1,2,\ldots,k-1)}},\frac{\phi_{(1,2,\ldots,k-1,j)}}{\phi_{(1,2,\ldots,k-1)}}\right].$

By well-known properties of the Wronskian operator, we finally have

$\varphi_{k+1,j}=\frac{1}{\varphi_{k,k}}\frac{\operatorname{\operatorname{Wr}}[\phi_{1,2,\ldots,k-1,k},\phi_{(1,2,\ldots,k-1,j)}]}{\left(\phi_{(1,2,\ldots,k-1)}\right)^{2}}=\frac{\phi_{(1,2,\ldots,k-1)}\phi_{(1,2,\ldots,k,j)}}{\varphi_{kk}\left(\phi_{(1,2,\ldots,k-1)}\right)^{2}}=\frac{\phi_{(1,2,\ldots,k,j)}}{\phi_{(1,2,\ldots,k)}}.$

∎

The following lemma establishes the desired result in the restricted gauge
$b_{1}=\cdots=b_{n}=1$.

######  Lemma 2.10.

Under the same setting as in Theorem 2.1, let $\tilde{T}_{0}=T_{0}$ and
$\tilde{T}_{k}:=pD^{2}+\tilde{q}_{k}D+\tilde{r}_{k}$ for $k\in\\{1,\ldots
n\\}$, where

(19) $\displaystyle\tilde{q}_{k}$ $\displaystyle:=q_{0}+kp^{\prime}$ (20)
$\displaystyle\tilde{r}_{k}:$
$\displaystyle=r_{0}+kq_{0}^{\prime}+\frac{1}{2}k(k-1)p^{\prime\prime}+\upsilon_{k}p^{\prime}+2\upsilon_{k}^{\prime}p.$

Let $\tilde{A}_{k}$ be as in Lemma 2.9. Then

(21) $\tilde{A}_{k}\tilde{T}_{k-1}=\tilde{T}_{k}\tilde{A}_{k},\quad
k=1,\ldots,n.$

###### Proof.

By direct calculation, we find that

$\displaystyle\tilde{T}_{k}\tilde{A}_{k}$
$\displaystyle=(pD^{2}+\tilde{q}_{k}D+\tilde{r}_{k})(D-\tilde{w}_{k})$
$\displaystyle=pD^{3}+(\tilde{q}_{k}-p\tilde{w}_{k})D^{2}+(\tilde{r}_{k}-\tilde{q}_{k}\tilde{w}_{k}-2p\tilde{w}_{k}^{\prime})D-(\tilde{r}_{k}\tilde{w}_{k}+\tilde{q}_{k}\tilde{w}_{k}^{\prime}+p\tilde{w}_{k}^{\prime\prime})$
$\displaystyle\tilde{A}_{k}\tilde{T}_{k-1}$
$\displaystyle=(D-\tilde{w}_{k})(pD^{2}+\tilde{q}_{k-1}D+\tilde{r}_{k-1})$
$\displaystyle=pD^{3}+(\tilde{q}_{k-1}-p\tilde{w}_{k}+p^{\prime})D^{2}+\big{(}\tilde{r}_{k-1}-\tilde{q}_{k-1}\tilde{w}_{k}+\tilde{q}_{k-1}^{\prime}\big{)}D+\tilde{r}_{k-1}^{\prime}-\tilde{w}_{k}\tilde{r}_{k-1}$

Hence, by inspection of the coefficients, the desired intertwining relation is
equivalent to the following three relations:

(22) $\displaystyle\tilde{q}_{k}=\tilde{q}_{k-1}+p^{\prime}$ (23)
$\displaystyle\tilde{r}_{k}=\tilde{r}_{k-1}+\tilde{q}_{k-1}^{\prime}+\tilde{w}_{k}p^{\prime}+2p\tilde{w}_{k}^{\prime}$
(24)
$\displaystyle\tilde{r}_{k}\tilde{w}_{k}+\tilde{q}_{k}\tilde{w}_{k}^{\prime}+p\tilde{w}_{k}^{\prime\prime}=\tilde{r}_{k-1}\tilde{w}_{k}-\tilde{r}_{k-1}^{\prime}$

By inspection, (19) entails (22). Then, using (19), (20) and
$\tilde{w}_{k}=\upsilon_{k}-\upsilon_{k-1}$, we find that

$\displaystyle\tilde{r}_{k}-\tilde{r}_{k-1}-\tilde{q}_{k-1}^{\prime}$
$\displaystyle=q_{0}^{\prime}+(k-1)p^{\prime\prime}+(\upsilon_{k}-\upsilon_{k-1})p^{\prime}+2(\upsilon_{k}^{\prime}-\upsilon_{k-1}^{\prime})p-(q^{\prime}_{0}+(k-1)p^{\prime\prime})$
$\displaystyle=\tilde{w}_{k}p^{\prime}+2\tilde{w}_{k}^{\prime}p,$

which establishes (23). Using (22) and (23), we can rewrite (24) as

$\displaystyle 0$
$\displaystyle=(\tilde{r}_{k}-\tilde{r}_{k-1})\tilde{w}_{k}+\tilde{q}_{k}\tilde{w}_{k}^{\prime}+p\tilde{w}_{k}^{\prime\prime}+\tilde{r}_{k-1}^{\prime}$
$\displaystyle=(\tilde{q}_{k-1}^{\prime}+p^{\prime}\tilde{w}_{k}+2p\tilde{w}_{k}^{\prime})\tilde{w}_{k}+(\tilde{q}_{k-1}+p^{\prime})\tilde{w}_{k}^{\prime}+p\tilde{w}_{k}^{\prime\prime}+\tilde{r}_{k-1}^{\prime}$
$\displaystyle=\big{(}p(\tilde{w}_{k}^{\prime}+\tilde{w}_{k}^{2})+\tilde{q}_{k-1}\tilde{w}_{k}+\tilde{r}_{k-1}\big{)}^{\prime}$

Let $\tilde{\psi}_{k}$ denote

$\tilde{\psi}_{k}=\varphi_{k,k}=\frac{\phi_{(1,\ldots,k)}}{\phi_{(1,\ldots,k-1)}},\quad\tilde{w}_{k}=(\log\tilde{\psi}_{k})^{\prime},$

and observe that

$p(\tilde{w}_{k}^{\prime}+\tilde{w}_{k}^{2})+\tilde{q}_{k-1}\tilde{w}_{k}+\tilde{r}_{k-1}=\frac{\tilde{T}_{k-1}\tilde{\psi}_{k}}{\tilde{\psi}_{k}},$

so (24) is equivalent to establishing

$\tilde{T}_{k-1}\tilde{\psi}_{k}=\lambda_{k}\tilde{\psi}_{k},\qquad
k\in\\{1,\dots,n\\}.$

The rest of the proof follows by induction. The base case holds because
$\tilde{\psi}_{1}=\phi_{1}$ and by assumption,
$T_{0}\phi_{1}=\lambda_{1}\phi_{1}$. Now, suppose that we have established
(21) for $j=1,\ldots,k-1$. By Lemma 2.9 and the inductive hypothesis,

$\displaystyle\tilde{T}_{k-1}\tilde{\psi}_{k}$
$\displaystyle=\tilde{T}_{k-1}\tilde{A}_{k-1}\cdots\tilde{A}_{1}\phi_{k}=\tilde{A}_{k-1}\tilde{T}_{k-2}\tilde{A}_{k-2}\cdots\tilde{A}_{1}\phi_{k}$
$\displaystyle=\tilde{A}_{k-1}\cdots\tilde{A}_{1}T_{0}\phi_{k}=\lambda_{k}\tilde{A}_{k-1}\cdots\tilde{A}_{1}\phi_{k}=\lambda_{k}\tilde{\psi}_{k}.$

∎

The next lemma shows the gauge transformation that connects $T_{k}$ with
$\tilde{T}_{k}$.

######  Lemma 2.11.

Let $T_{k}$ be as in Theorem 2.1 and $\tilde{T}_{k}$ be as in Lemma 2.10.
Setting $s_{k}:=b_{1}\dots b_{k}$, we have

(25) $T_{k}=s_{k}\tilde{T}_{k}s^{-1}_{k},\quad k=1,\ldots,n.$

###### Proof.

Observe that $\sigma_{k}=(\log s_{k})^{\prime}$. By direct calculation,

$\displaystyle s_{k}Ds_{k}^{-1}$ $\displaystyle=D-\sigma_{k},$ $\displaystyle
s_{k}D^{2}s_{k}^{-1}$
$\displaystyle=(s_{k}Ds_{k}^{-1})^{2}=(D-\sigma_{k})^{2}=D^{2}-2\sigma_{k}D+\sigma_{k}^{2}-\sigma_{k}^{\prime},$

Hence,

$\displaystyle q_{k}$ $\displaystyle=\tilde{q}_{k}-2\sigma_{k}p,$
$\displaystyle r_{k}$
$\displaystyle=\tilde{r}_{k}+(\sigma_{k}^{2}-\sigma_{k}^{\prime})p-\sigma_{k}\tilde{q}_{k},$

as was to be shown. ∎

With all the previous elements, proving Theorem 2.1 for general factorization
gauges becomes a straightforward computation

###### Proof of Theorem 2.1.

We observe that $A_{k}$ is related with $\tilde{A}_{k}$ in Lemma 2.9 by
$s_{k}=b_{1}\dots b_{k}$ as follows:

$s_{k}\tilde{A}_{k}s_{k}^{-1}=D-\tilde{w}_{k}-\sigma_{k}=A_{k}.$

Hence, by Lemma 2.11,

$T_{k}A_{k}=s_{k}\tilde{T}_{k}\tilde{A}_{k}s_{k}^{-1}=s_{k}\tilde{A}_{k}\tilde{T}_{k-1}s_{k}^{-1}=A_{k}T_{k-1},$

as was to be shown. ∎

######  Remark 2.12.

In light of Lemma 2.11, a different choice of $b_{1},\ldots,b_{n}$ results in
a gauge transformation of the operators $T_{1},\ldots,T_{n}$. It is for this
reason that we refer to $b_{1},\ldots,b_{n}$ as factorization gauges.
Moreover, by (LABEL:eq:qkrk), the coefficients of $T_{n}$ are defined directly
in terms of $\sigma_{n}=(\log s_{n})^{\prime}$ and $\upsilon_{n}$. The
Wronskian $\phi_{(1,2,\ldots,n)}$ is alternating in its indices and its log-
derivative $\upsilon_{n}$ is invariant with respect to permutations of the set
$\\{1,2,\ldots,n\\}$. Thus, we see that $T_{n}$ depends only on the choice of
the seed eigenfunctions $\phi_{1},\ldots,\phi_{n}$ — irrespective of their
order — and on the product of the factorization gauges $s_{n}=b_{1}\cdots
b_{n}$. Fixing the seed eigenfunctions, but choosing a different $s_{n}$
amounts to a gauge transformation of $T_{n}$.

##  3\. Confluent Darboux transformations

In this section we generalize the concept of Darboux transformations to allow
for repeated eigenvalues. Notice that the construction in Section 2 fails if
the eigenvalues of the factorization eigenfunctions are not all distinct,
because then some of the seed functions may not be linearly independent, which
leads to the vanishing of the Wronskians in the denominator of (14). In order
to allow for repeated eigenvalues, we will allow some of our seed
eigenfunctions to become generalized eigenfunctions.

######  Definition 3.1.

We say that two second-order rational operators $T_{0}$ and $T_{2}$ are
connected by a _confluent Darboux transformation_ if there exists a second-
order rational operator $T_{1}$ such that $T_{0}\to T_{1}\to T_{2}$ is a
2-step Darboux transformation and the corresponding eigenvalues, as defined in
Proposition 2.2, satisfy $\lambda_{1}=\lambda_{2}$.

As we shall see, the factorization eigenfunction for the second step $T_{1}\to
T_{2}$ will not be related to an eigenfunction of $T_{0}$ but to a generalized
eigenfunction, which motivates the following definition.

######  Definition 3.2.

Let $T$ be a linear differential operator. We say that $\phi$ is an
_$n^{\text{th}}$ order generalized eigenfunction_ of $T$ if
$(T-\lambda)^{n}\phi=0$, but $(T-\lambda)^{n-1}\phi\neq 0$.

For our purposes it will be sufficient to use only first-order generalized
eigenfunctions; however, the more general construction can be found in [6]. We
now show that a CDT (confluent Darboux transformation) can be generated by a
seed eigenfunction and a corresponding second-order generalized eigenfunction.

In order to build a CDT, we start with a second-order rational operator
$T_{0}=pD^{2}+q_{0}D+r_{0}$, a quasi-rational seed eigenfunction $\phi$ with
eigenvalue $\lambda$ and rational factorization gauges $b_{1}$ and $b_{2}$.
Let $T_{0}\to T_{1}$ be a 1-step Darboux transformation with factorization
function $\phi$ at factorization eigenvalue $\lambda$. We now wish to perform
a second Darboux transformation on $T_{1}$ using the repeated eigenvalue
$\lambda$ and factorization gauge $b_{2}$. As shown below, this requires that
the second seed function be a generalized eigenfunction of $T_{0}$. We
therefore seek to construct a function $\phi^{(1)}$ such that
$(T_{0}-\lambda)\phi^{(1)}=\phi$. The following lemma shows how to achieve
this.

######  Lemma 3.3.

Let $T_{0}=pD^{2}+q_{0}D+r_{0}$ and let $\phi$ an eigenfunction of $T_{0}$
with eigenvalue $\lambda$, i.e. $T_{0}\phi=\lambda\phi$. Then the particular
solution of the inhomogeneous equation $(T_{0}-\lambda)y=\phi$ is given by

(26) $\displaystyle\phi^{(1)}(z)$
$\displaystyle=\phi(z)\int^{z}\left(\frac{\mu(u)}{\phi(u)^{2}}\int^{u}\frac{\phi^{2}(s)}{p(s)\mu(s)}ds\right)du,$

where

(27) $\mu(z):=\exp\left(-\int^{z}\frac{q_{0}(u)}{p(u)}du\right),$

Also, a linearly independent solution of the homogenous equation
$(T_{0}-\lambda)y=0$ is given by

(28) $\phi^{\perp}(z):=\phi(z)\int^{z}\frac{\mu(u)}{\phi(u)^{2}}du.$

######  Remark 3.4.

Having fixed $\mu$ as in (27), the above definition of $\phi^{(1)}$
incorporates two additional constants of integration which correspond to
linear combination of the solutions of the homogeneous equation, $\phi$ and
$\phi^{\perp}$. We will fix one of these constants once we consider an
explicit form for $p$, by imposing an appropriate lower bound for the
integral. The other constant of integration serves as a natural deformation
parameter in the CDT construction. This will be explained in more detail later
on.

###### Proof of Lemma 3.3.

We first consider a complementary solution to the eigenvalue equation
$T_{0}y=\lambda y$, which we will denote $\phi^{\perp}$. We can obtain an
explicit formula for $\phi^{\perp}$ via reduction of order. Substituting the
form $\phi^{\perp}:=f\phi$ into the equation
$T\phi^{\perp}=\lambda\phi^{\perp}$ yields the equation

$\displaystyle\frac{f^{\prime\prime}}{f^{\prime}}=-\frac{q_{0}}{p}-2w,\quad\mbox{
with }w=\frac{\phi^{\prime}}{\phi}.$

Noticing that $\frac{\mu^{\prime}}{\mu}=-\frac{q_{0}}{p}$, it follows that

$\displaystyle f(z)=\int^{z}\frac{\mu(u)}{\phi(u)^{2}}du.$

The above definitions are purely formal in that we have not specified the
lower bound of the integrals. This means that $\phi^{\perp}$ is defined up to
a constant multiple of $\phi$, and that $\mu$ is defined up to a choice of
positive multiplicative constant.

We now construct $\phi^{(1)}$ using variation of parameters. We set

(29) $\phi^{(1)}:=\hat{\rho}\phi+\rho\phi^{\perp},$

where $\hat{\rho}$ and $\rho$ are unknown functions satisfying

$\displaystyle\hat{\rho}{}^{\prime}\phi+\rho^{\prime}\phi^{\perp}=0.$

Since $(T_{0}-\lambda)\phi^{(1)}=\phi$, then they also satisfy

$\displaystyle
p\big{(}\hat{\rho}^{\prime\prime}\phi+2\hat{\rho}^{\prime}\phi^{\prime}+\rho^{\prime\prime}\phi^{\perp}+2\rho^{\prime}(\phi^{\perp})^{\prime}\big{)}=\hat{\rho}\phi+\rho\phi^{\perp}.$

Solving for the functions $\hat{\rho}$ and $\rho$ satisfying the above system
of equations, we find that

$\displaystyle\hat{\rho}(z)$
$\displaystyle=-\int^{z}\phi(u)\phi^{\perp}(u)W(u)du,$ (30)
$\displaystyle\rho(z)$ $\displaystyle=\int^{z}\phi^{2}(u)W(u)du,$

where

(31)
$W(z):=\big{(}p(z)\mu(z)\big{)}^{-1}=p(z)^{-1}\exp\left(\int^{z}\frac{q_{0}(u)}{p(u)}du\right).$

Now integrating by parts it follows that we can express $\phi^{(1)}$ as

(32) $\displaystyle\phi^{(1)}(z)$
$\displaystyle=\phi(z)(f(z)\rho(z)+\hat{\rho}(z))=\phi(z)\left(f(z)\rho(z)-\int^{z}f(u)\rho^{\prime}(u)du\right)$
$\displaystyle=\phi(z)\int^{z}\rho(u)f^{\prime}(u)du=\phi(z)\int^{z}\frac{\rho(u)\mu(u)}{\phi(u)^{2}}du.$

∎

After the first Darboux transformation $T_{0}\to T_{1}$ at factorization
eigenvalue $\lambda$, we define the intertwiner $A_{1}=b_{1}(D-w_{1})$, with
$w_{1}=(\log\phi)^{\prime}$. The first candidate for factorization function
for the second Darboux transformation $T_{1}\to T_{2}$ would be the image of
$\phi^{\perp}$ under $A_{1}$:

(33) $\psi^{\perp}:=A_{1}\phi^{\perp}=b_{1}\frac{\mu}{\phi}.$

Indeed, since $T_{1}\psi^{\perp}=\lambda\psi^{\perp}$, we could employ
$\psi^{\perp}$ as a factorization eigenfunction for a 1-step Darboux
transformation on $T_{1}$. However, this choice of eigenfunction produces the
_inverse Darboux transformation_ $T_{1}\to T_{0}$.

In order to construct an operator $T_{2}$ distinct from $T_{0}$, we need to
consider another candidate: the image of the generalized eigenfunction
$\phi^{(1)}$ under $A_{1}$. Indeed, we define the factorization eigenfunction
for $T_{1}$ to be

(34) $\psi_{2}:=A_{1}\phi^{(1)}=\rho\psi^{\perp},$

with $\rho$ as in (30). The second equality is true because

$\operatorname{\operatorname{Wr}}[\phi,\phi^{(1)}]=\operatorname{\operatorname{Wr}}\left[\phi,\phi\int^{z}\frac{\rho\mu}{\phi^{2}}\right]=\rho\mu$

and therefore

$\psi_{2}=\frac{b_{1}}{\phi}\operatorname{\operatorname{Wr}}[\phi,\phi^{(1)}]=\frac{b_{1}}{\phi}\rho\mu=\rho\psi^{\perp}.$

The key observation is that although $\phi^{(1)}$ is only a generalized
eigenfunction of $T_{0}$, its image $\psi_{2}$ is a true eigenfunction of
$T_{1}$ at eigenvalue $lambda$:

(35)
$(T_{1}-\lambda)\psi_{2}=(T_{1}-\lambda)A_{1}\phi^{(1)}=A_{1}(T_{0}-\lambda)\phi^{(1)}=A_{1}\phi_{1}=0,$

and thus it can be employed for the second Darboux transformation $T_{1}\to
T_{2}$. We summarize the construction of a confluent Darboux transformation
$T_{0}\to T_{2}$ with a generalized eigenfunction in the following
proposition.

######  Proposition 3.5.

Let $T_{0}=pD^{2}+q_{0}D+r_{0}$ be a second-order rational operator, $\phi$ a
quasi-rational eigenfunction of $T_{0}$ with eigenvalue $\lambda$, and
$b_{1},b_{2}$ a choice of non-zero rational functions. Let $\mu,\rho$ be
defined as per (27), (30), respectively, and assume that $\rho$ is a quasi-
rational function. Let $T_{2}=pD^{2}+q_{2}D+r_{2}$, where $p_{2}$ and $r_{2}$
are defined in (LABEL:eq:qkrk) with

$\upsilon_{1}:=(\log\phi)^{\prime},\quad\upsilon_{2}:=\big{(}\log(\rho\mu)\big{)}^{\prime}.$

Then $T_{0}$ and $T_{2}$ are connected by a confluent Darboux transformation

###### Proof.

Firstly, we observe that $T_{2}$ is a second-order rational operator because
of the assumption on $\mu$. Then, it remains to show there is $T_{1}$ such
that $T_{0}\to T_{1}\to T_{2}$ is a 2-step DT with $\lambda_{1}=\lambda_{2}$.
Defining $T_{1},A_{1},A_{2}$ as per (13)-(LABEL:eq:wkgen), with
$b_{j},\upsilon_{j}$, $j\in\\{1,2\\}$ as in the statement, the result then
follows. ∎

######  Remark 3.6.

Observe that the construction of a confluent Darboux transformation amounts to
applying the extended Crum Wronskian formula derived in Theorem 2.1 for $k=2$
with the only modification that one of the seed functions of $T_{0}$ is a true
eigenfunction, but the second one is a generalized eigenfunction, i.e. apply
the formulas in Theorem 2.1 with

$\phi_{1}=\phi,\quad\phi_{2}=\phi^{(1)},\quad\lambda_{1}=\lambda_{2}=\lambda.$

######  Remark 3.7.

The definition (26) of the second seed function $\phi^{(1)}$ involves two
indefinite integrals. The outer integral provides no extra freedom, because it
means that $\phi^{(1)}$ is defined up to an additive term $C\phi$ but this
term will vanish in the Wronskian $\phi_{(1,2)}$. The lower bound of the inner
integral, however, introduces a free real parameter, which is a characteristic
feature of the CDT.

######  Remark 3.8.

The assumption that $\frac{\rho^{\prime}}{\rho}$ be rational ensures that
$\upsilon_{2}$, as defined above, is rational. This assumption may be restated
as the condition that $\phi^{2}W$, the integrand of (30), is a rational
function with vanishing residues. Verifying the rationality of the CDT is key
to ensure that the transformed operator has polynomial eigenfunctions.

######  Remark 3.9.

The confluent aspect of a CDT comes from a conceptual formula for the
generalized eigenvalue equation $(T_{0}-\lambda)\phi^{(1)}=\phi$. We note
that, despite not being made explicit, the seed eigenfunction $\phi$ depends
on the eigenvalue $\lambda$. This dependence can be recovered by imposing
initial conditions on $\phi$ and $\phi^{\prime}$. Thus, starting from the
eigenvalue equation $(T_{0}-\lambda)\phi=0$, we can differentiate with respect
to $\lambda$ to find that

(36)
$\displaystyle(T_{0}-\lambda)\left[\frac{\partial\phi}{\partial\lambda}\right]=\phi.$

This equation implies a rather simple formula for defining $\phi^{(1)}$, which
is

(37) $\displaystyle\phi^{(1)}=\frac{\partial\phi}{\partial\lambda}.$

This formula is not of much practical use, since the functional dependence on
$\lambda$ is typically impossible to express explicitly. However, this
expression is of conceptual importance and the derivative can be seen as a
limiting case of the ordinary Darboux transformation, where the eigenvalues
converge as $\lambda_{2}\to\lambda_{1}$. Hence the name _confluent_ Darboux
transformation.

In this section we have seen how to build a confluent Darboux transformation
as a 2-step Darboux transformation with a generalized eigenfunction,
introducing in the process a free real parameter. This construction can be
iterated at different eigenvalues to create chains of CDTs: perform a CDT on
the first operator $T_{0}$ at eigenvalue $\lambda_{1}$, which is followed by a
CDT on $T_{2}$ at eigenvalue $\lambda_{2}\neq\lambda_{1}$, which yields the
operator $T_{4}$, etc.

Chains of operators may be constructed through an arbitrary finite number of
CDTs in this fashion, thereby leading to an operator with an arbitrary number
of free real parameters, which is Darboux connected to the original $T_{0}$.
In the following section we show how to construct a CDT chain starting on the
Gegenbauer polynomials, and leading to the exceptional Gegenbauer polynomials
of the second kind.

##  4\. Exceptional Gegenbauer operators and polynomials

In this Section we apply the theory developed in Sections 2 and 3 to construct
a chain of confluent Darboux transformations on the classical Gegenbauer
operator.

###  4.1. Definition of exceptional Gegenbauer operators and polynomials

In [12] it was shown that an exceptional operator in the Hermite, Laguerre or
Jacobi class must have a very specific form. We define in this section
exceptional Gegenbauer polynomials and operators attending to this particular
form (as a particular class of exceptional Jacobi operators), postponing for
later sections the discussion on how the construction of specific families is
achieved.

######  Definition 4.1.

Let $\tau=\tau(z)$ be a non-zero polynomial and $\alpha\in\mathbb{R}$. We say
that the differential expression

(38)
$T^{(\alpha)}_{\tau}(z,D_{z}):=(1-z^{2})\left(D_{z}^{2}-2\frac{\tau_{z}}{\tau}D_{z}+\frac{\tau_{zz}}{\tau}\right)-(2\alpha+1)zD_{z}+(2\alpha-1)z\frac{\tau_{z}}{\tau}.$

is an _exceptional Gegenbauer operator_ if $T^{(\alpha)}_{\tau}$ admits
eigenpolynomials $\\{\pi_{i}(z)\\}_{i\in\mathbb{N}_{0}}$ such that the degree
sequence $\\{\deg\pi_{i}\\}_{i\in\mathbb{N}_{0}}$ is missing finitely many
“exceptional” degrees.

In the above definition, it should be stressed that only very specific
polynomials $\tau(z)$ in (38) will lead to $T^{(\alpha)}_{\tau}$ being an
exceptional Gegenbauer operator, i.e. having an infinite number of polynomial
eigenfunctions. The following sections are devoted to describing a class of
polynomials $\tau$ obtained by applying a multi-step CDT on $\tau=1$, which
ensures that this is indeed the case. Note that there is no restriction on the
parameter $\alpha$ at this stage. In the following section we will see that
$\alpha$ must be a half-integer for the Confluent Darboux transformations to
be rational. This means that for standard Darboux transformations (see Section
2) the parameter $\alpha$ can be real, leading to $X$-Gegenbauer polynomials
of the first kind, but for confluent Darboux transformations (see Section 3)
the parameter $\alpha\in\mathbb{N}_{0}+1/2$, leading to excecptional
Gegenbauer polynomials of the second kind.

Observe also, by contrast to classical orthogonal polynomials, that we _are
not_ assuming that $\deg\pi_{i}=i$. Furthermore, without loss of generality,
it will be convenient to assume that

$\displaystyle\deg\pi_{i}\neq\deg\pi_{j}\;\mbox{ if }i\neq j.$

As usual, we speak of exceptional Gegenbauer polynomials when the
eigenpolynomials $\\{\pi_{i}(z)\\}_{i\in\mathbb{N}_{0}}$ define a complete
orthogonal polynomial system.

######  Definition 4.2.

Let $\tau(z)$ be a polynomial and $\alpha\in\mathbb{R}$ . We say that the set
$\\{\pi_{i}(z)\\}_{i\in\mathbb{N}_{0}}$ is a family of _exceptional Gegenbauer
polynomials_ with weight

(39)
$W^{(\alpha)}_{\tau}(z):=\frac{(1-z^{2})^{\alpha-\frac{1}{2}}}{\tau(z)^{2}},$

if the following conditions hold:

  1. (a)

$\tau(z)$ does not vanish on $I=[-1,1]$;

  2. (b)

$\\{\pi_{i}(z)\\}_{i\in\mathbb{N}_{0}}$ are eigenpolynomials of an
$X$-Gegenbauer operator (38);

  3. (c)

The polynomials $\\{\pi_{i}(z)\\}_{i\in\mathbb{N}_{0}}$ form a complete set in
the Hilbert space $\mathrm{L}^{2}(I,W^{(\alpha)}_{\tau})$.

Note that there is no need to include an explicit orthogonality assumption in
the above definition, because orthogonality of the eigenpolynomials follows
from assumptions (a) and (b). Also note that there is no need for
supplementary assumptions regarding the corresponding eigenvalues as these are
necessarily quadratic functions of the degree sequence. More specifically, we
establish these results in the next two lemmas.

######  Lemma 4.3.

Let $T^{(\alpha)}_{\tau}$ be an exceptional Gegenbauer operator and let
$\\{\pi_{i}\\}_{i\in\mathbb{N}_{0}}$ be its associated eigenpolynomials, i.e.

(40) $T^{(\alpha)}_{\tau}\pi_{i}=\lambda_{i}\pi_{i}$

Then, necessarily

(41) $\displaystyle\lambda_{i}$
$\displaystyle=-d_{i}(2\alpha+d_{i}),\quad\text{where
}d_{i}=\deg\pi_{i}-\deg\tau$

###### Proof.

Let $\tau(z)$ be a polynomial and $\pi_{i}(z)$ an eigenpolynomial of
$T^{(\alpha)}_{\tau}$ with eigenvalue $\lambda_{i}$. Expliciltly, by (38), we
have

(42)
$(1-z^{2})\left(\tau\pi_{i}^{\prime\prime}-2\tau^{\prime}\pi_{i}^{\prime}+\tau^{\prime\prime}\pi_{i}\right)-(2\alpha+1)z\tau\pi_{i}^{\prime}+(2\alpha-1)z\tau^{\prime}\pi_{i}=\lambda_{i}\tau\pi_{i}.$

Let $n=\deg\pi_{i}$ and $m=\deg\tau$ and without loss of generality suppose
that both $\tau(z),\pi_{i}(z)$ are monic; i.e. $\tau(z)=z^{m}+\ldots$ and
$\pi(z)=z^{n}+\ldots$. Notice that the highest power of $z$ on either side of
(42) is $m+n$. Hence, for the above equation to hold, the two coefficients on
$z^{m+n}$ must be equal. Considering only the highest power of $z$ in each
term, yields the equation

$\displaystyle\lambda_{i}z^{m+n}=\left(-n(n-1)+2mn-m(m-1)-(2\alpha+1)n+(2\alpha-1)m\right)z^{m+n}.$

Hence

$\displaystyle\lambda_{i}$
$\displaystyle=-n(n-1)+2mn-m(m-1)-(2\alpha+1)n+(2\alpha-1)m$
$\displaystyle=-n^{2}+2mn-m^{2}-2\alpha n+2\alpha m$
$\displaystyle=-(n-m)^{2}-2\alpha(n-m)$ $\displaystyle=-(n-m)(2\alpha+n-m),$

as was to be shown. ∎

######  Lemma 4.4.

A family of exceptional Gegenbauer polynomials
$\\{\pi_{i}\\}_{i\in\mathbb{N}_{0}}$ is necessarily orthogonal with respect to
the corresponding weight (39):

(43) $\int_{I}\pi_{i}(z)\pi_{j}(z)W^{(\alpha)}_{\tau}(z)dz=0\;\mbox{ if }i\neq
j.$

###### Proof.

Multiplying the eigenvalue equation $T^{(\alpha)}_{\tau}y=\lambda y$ by
$W^{(\alpha)}_{\tau}$ yields a Sturm-Liouville eigenvalue equation

(44)
$(P^{(\alpha)}_{\tau}y^{\prime})^{\prime}+R^{(\alpha)}_{\tau}y=W^{(\alpha)}_{\tau}\lambda
y,$

where

$\displaystyle P^{(\alpha)}$
$\displaystyle:=(1-z^{2})^{\alpha+\frac{1}{2}}\tau^{-2}$ $\displaystyle
R^{(\alpha)}$
$\displaystyle:=(1-z^{2})^{\alpha+\frac{1}{2}}\tau_{zz}\tau^{-3}+(2\alpha-1)z(1-z^{2})^{\alpha-\frac{1}{2}}\tau_{z}\tau^{-3}$

Lagrange’s identity now gives

(45)
$\int\left(y_{1}T^{(\alpha)}_{\tau}y_{2}-y_{2}T^{(\alpha)}_{\tau}y_{1}\right)W^{(\alpha)}_{\tau}=P^{(\alpha)}_{\tau}\operatorname{\operatorname{Wr}}[y_{1},y_{2}].$

Since the eigenvalues $\lambda_{i}$ are distinct, assumption (a) and (45)
imply that the eigenpolynomials satisfiy orthogonality relations. ∎

The base case of the class of exceptional Gegenbauer operators is the
classical Gegenbauer operator $T^{(\alpha)}=T^{(\alpha)}_{\tau_{0}}$, where
$\tau_{0}(z)=1$. In this case the general form (38) simplifies to (1) and the
eigenpolynomials are the classical Gegenbauer polynomials [2]

(46) $C^{(\alpha)}_{i}:=\sum_{k=0}^{\lfloor
i/2\rfloor}(-1)^{k}\frac{\Gamma(i-k+\alpha)}{\Gamma(\alpha)k!(i-2k)!}(2z)^{i-2k}.$

These classical orthogonal polynomials do have $\deg C^{(\alpha)}_{i}=i$, and
they satisfy the eigenvalue relation

$\displaystyle T^{(\alpha)}C^{(\alpha)}_{i}=\lambda_{i}C^{(\alpha)}_{i},\quad
i\in\mathbb{N}_{0},$

with $\lambda_{i}=-i(2\alpha+i)$. If $\alpha>-\frac{1}{2}$ the polynomials
$\\{C^{(\alpha)}_{i}\\}_{i\in\mathbb{N}_{0}}$ form a complete set in
$\mathrm{L}^{2}(I,W^{(\alpha)})$ and they satisfy the orthogonality relation

$\int_{I}C^{(\alpha)}_{i}(u)C^{(\alpha)}_{j}(u)W^{(\alpha)}(u)du=\nu^{(\alpha)}_{i}\delta_{ij},\quad
i,j\in\mathbb{N}_{0}$

where

(47) $\displaystyle W^{(\alpha)}(z)$
$\displaystyle:=W^{(\alpha)}_{\tau_{0}}(z)=(1-z^{2})^{\alpha-\frac{1}{2}};$
(48) $\displaystyle\nu^{(\alpha)}_{i}$ $\displaystyle:=\frac{\pi
2^{1-2\alpha}\Gamma(i+2\alpha)}{i!(i+\alpha)\Gamma(\alpha)^{2}},\quad
i\in\mathbb{N}_{0}.$

We can now apply the theory of CDTs to generate exceptional orthogonal
polynomials.

Once the class of exceptional Gegenbauer operators and polynomials has been
defined, we will show that this class is not empty by describing in the next
sections the construction of exceptional Gegenbauer polynomials of the second
kind via a sequence of confluent Darboux transforamtions.

###  4.2. Factorizations of exceptional Gegenbauer operators

Let us start by describing the factorization of an $X$-Gegenbauer operator
(38) in order to define a $1$-step Darboux transformation. After that, we will
combine two Darboux transformations to define a CDT.

Given rational functions $\tau(z)$ and $\pi(z)$, and a real constant $\alpha$,
we define the following two first-order rational operators:

(49) $\displaystyle A_{\tau\pi}(z,D_{z})$
$\displaystyle:=\tau(z)^{-1}(\pi(z)D_{z}-\pi^{\prime}(z)),$ $\displaystyle
B^{(\alpha)}_{\pi\tau}(z,D_{z})$
$\displaystyle:=(1-z^{2})A_{\pi\tau}(z,D_{z})-(2\alpha+1)z\tau(z)\pi(z)^{-1}.$

It will be useful to express the above operators in terms of Wronskian
determinants by setting

(50) $\displaystyle\hat{\pi}(z)$
$\displaystyle:=(1-z^{2})^{-\alpha-\frac{3}{2}}\pi(z),$
$\displaystyle\hat{\tau}(z)$
$\displaystyle:=(1-z^{2})^{-\alpha-\frac{1}{2}}\tau(z),$

so that

$\displaystyle A_{\tau\pi}y=\tau^{-1}\operatorname{\operatorname{Wr}}[\pi,y],$
$\displaystyle
B^{(\alpha)}_{\pi\tau}y=\hat{\pi}^{-1}\operatorname{\operatorname{Wr}}[\hat{\tau},y].$

Observe that it would be sufficient to define only $A_{\tau\pi}$ since the two
operators are related by

(51) $\displaystyle B^{(\alpha)}_{\pi\tau}=A_{\hat{\pi}\hat{\tau}}.$

However, defining both operators separately will make for simpler notation
going forward. With these first order operators, we can now describe the
factorization of an $X$-Gegenabuer operator (38) in the following proposition.

######  Proposition 4.5.

Let $\alpha\in\mathbb{R}$, $\tau(z)$ be a polynomial, and
$T^{(\alpha)}_{\tau}$ be an exceptional Gegenabuer operator as in (38). Assume
that $\pi(z)$ is an eigenpolynomial of $T^{(\alpha)}_{\tau}$ with eigenvalue
$\lambda$, i.e. $T^{(\alpha)}_{\tau}\pi=\lambda\pi$ . We then have the
following factorizations:

(52) $\displaystyle B^{(\alpha)}_{\pi\tau}A_{\tau\pi}$
$\displaystyle=T^{(\alpha)}_{\tau}-\lambda,$ $\displaystyle
A_{\tau\pi}B^{(\alpha)}_{\pi\tau}$
$\displaystyle=T^{(\alpha+1)}_{\pi}-\hat{\lambda},$

where

(53) $\hat{\lambda}=\lambda+2\alpha+1.$

###### Proof.

The proof follows by a straightforward computation. ∎

We say that the transformation $T^{(\alpha)}_{\tau}\to
T^{(\alpha+1)}_{\pi}-(2\alpha+1)$ is a formally _state-deleting_ Darboux
transformation, because the second operator no longer has an eigenpolynomial
at the eigenvalue $\lambda$. Likewise, we refer to the transformation
$T^{(\alpha+1)}_{\pi}\to T^{(\alpha)}_{\tau}{+(2\alpha+1)}$ as a formally
_state-adding_ Darboux transformation, because the operator
$T^{(\alpha)}_{\tau}$ gains an extra polynomial eigenfunction $\pi$ at
eigenvalue $\lambda$. The following proposition describes the factorization
eigenfunction for this dual factorization.

######  Proposition 4.6.

Let $\alpha\in\mathbb{R}$, $\tau(z)$ and $\pi(z)$ be non-zero polynomials and
let $\hat{\tau}$ be as in (50). Suppose that the following eigenvalue equation
holds: $T^{(\alpha+1)}_{\pi}\hat{\tau}=\hat{\lambda}\hat{\tau}$. Then the
factorizations (52) also hold.

###### Proof.

By direct calculation, we obtain that

(54) $\pi
T_{\pi}^{(\alpha+1)}\hat{\tau}=\hat{\tau}T_{\tau}^{(\alpha)}\pi+(2\alpha+1)\pi\hat{\tau}.$

Hence, $T^{(\alpha)}_{\tau}\pi=\lambda\pi$ is equivalent to
$T^{(\alpha+1)}_{\pi}\hat{\tau}=\hat{\lambda}\hat{\tau}$. This, together with
(51), implies the result.

∎

###  4.3. Confluent Darboux transformations of exceptional Gegenbauer
operators

In this section, we apply the theory of confluent Darboux transformations
developed in Section 3 to exceptional Gegenbauer operators. With the
factorizations introduced in the previous section, we will realize a CDT of an
exceptional Gegenbauer operator $T_{0}$ by a state-deleting transformation
$T_{0}\to T_{1}$ followed by a 1-parameter family of state-adding
transformations $T_{1}\to T_{2}$ at the same eigenvalue. Next we derive
certain recursive formulas that connect the $\tau$-functions and
eigenpolynomials of two $X$-Gegenbauer operators connected by a CDT.

Starting from the classical Gegenbauer operator we will construct a chain of
CDTs and a recursive construction of exceptional Gegenbauer operators of
second kind. This recursive construction will be discussed in more detail in
the following section.

######  Remark 4.7.

While the definitions of X-Gegenbauer operators and their factorizations in
the previous Section hold for any $\alpha\in\mathbb{R}$, we need to assume
that certain integrals like (55) define rational functions, which requires
that $\alpha\in\mathbb{N}_{0}+\frac{1}{2}$ from here on.

Suppose moreover that $\tau(z)$ is a polynomial and $T^{(\alpha)}_{\tau}$ is
an exceptional Gegenbauer operator, as per (38), with eigenpolynomials
$\\{\pi_{i}\\}_{i\in\mathbb{N}_{0}}$. We define the functions

(55)
$\displaystyle\rho_{ij}(z):=\int_{-1}^{z}\pi_{i}(u)\pi_{j}(u)W^{(\alpha)}_{\tau}(u)du,\quad
i,j\in\mathbb{N}_{0};$ (56)
$\displaystyle\tau_{m}(z,t):=\tau(z)(1+t\rho_{mm}(z)),\quad
m\in\mathbb{N}_{0};$ (57)
$\displaystyle\pi_{m;i}(z,t):=(1+t\rho_{mm}(z))\pi_{i}(z)-t\rho_{im}(z)\pi_{m}(z),\quad
i,m\in\mathbb{N}_{0},$

where the integral in (55) denotes a formal anti-derivative that vanishes at
$z=-1$.

######  Remark 4.8.

Throughout this Section, we assume that $\tau_{m}(z)$ and $\pi_{m;i}(z,t)$ are
polynomials in $z$ and $\rho_{ij}(z)$ is a rational function of $z$. In
principle, this assumption seems a strong requirement when looking at (55) and
(39). However, we will see in Section 5 that whenever these quantities are
connected recursively to the classical Gegenbauer operator and polynomials,
there exist matrix formulas that establish the polynomial character of
$\tau_{m}(z)$ and $\pi_{m;i}(z,t)$ and the rational character of
$\rho_{ij}(z)$ by construction.

######  Proposition 4.9.

For a given $m\in\mathbb{N}_{0}$, the operators
$T^{(\alpha)}_{\tau},T^{(\alpha)}_{\tau_{m}}$ are related by a confluent
Darboux transformation generated by the seed functions
$\\{\pi_{m},{\color[rgb]{0,0,1}\pi_{m}^{(1)}\\}}$.

###### Proof.

Set $T_{0}=T^{(\alpha)}_{\tau}$, and recall that, by assumption,
$T_{0}\pi_{m}=\lambda_{m}\pi_{m}$. We set

$\displaystyle T_{1}:=T^{(\alpha+1)}_{\pi_{m}}-(2\alpha+1),\quad
T_{2}:=T^{(\alpha)}_{\tau_{m}},$ $\displaystyle A_{1}:=A_{\tau,\pi_{m}},\quad
A_{2}:=B^{(\alpha)}_{\pi_{m},\tau_{m}}$

By Proposition 4.5, $T_{0}\to T_{1}$ is a state-deleting Darboux
transformation. Consequently,

(58) $A_{1}T_{0}=T_{1}A_{1}.$

We now claim that the transformation $T_{1}\to T_{2}$ is a 1-parameter family
of state-adding Darboux transformations. By inspection of (38), we find that
the relevant coefficient functions of $T_{0}$ are

$\displaystyle p:=1-z^{2},\quad
q_{0}:=-2(1-z^{2})\frac{\tau_{z}}{\tau}-(2\alpha+1)z.$

Then the function $\mu(z)$ in (27) satisfies

$\displaystyle\frac{\mu^{\prime}}{\mu}=-\frac{q_{0}}{p}=2\frac{\tau_{z}}{\tau}+\frac{(2\alpha+1)z}{1-z^{2}},$

and hence

$\displaystyle\mu(z):=(1-z^{2})^{-\alpha-\frac{1}{2}}\tau(z)^{2}.$

Following (32), we construct a generalized eigenfunction of $T_{0}$ as shown
in Section 3. We define

(59)
$\displaystyle\pi_{m}^{(1)}(z;t):=\pi_{m}(z)\int_{-1}^{z}(1+t\rho_{mm}(s))\hat{\tau}(s)^{2}W_{\pi_{m}}^{(\alpha+1)}(s)ds,$
where $\displaystyle\hat{\tau}(z):=(1-z^{2})^{-\alpha-1/2}\tau(z)$

and $W_{\pi_{m}}^{(\alpha+1)}$ is given in (39). Observe that
$W^{(\alpha)}_{\tau}=(p\mu)^{-1}$, in agreement with (31). The definitions of
$\rho_{mm},\pi_{m}^{(1)},\hat{\tau}$ agree with the definitions of
$\rho,\phi^{(1)},\psi_{1}^{\perp}$ in (30), (32) and (33), respectively.
Consequently, as shown in Section 3, $\pi_{m}^{(1)}$ is a first-order
generalized eigenfunction of $T_{0}$, because it satisfies the equation

(60) $\displaystyle(T_{0}-\lambda_{m})\pi_{m}^{(1)}=t\pi_{m}.$

A direct calculation shows that

$\displaystyle A_{1}\pi_{m}^{(1)}=\hat{\tau}_{m}.$

where

(61)
$\hat{\tau}_{m}(z,t):=(1-z^{2})^{-\alpha-1/2}\tau_{m}(z,t)=\hat{\tau}(z)(1+t\rho_{mm}(z)).$

Hence, by (58),

$\displaystyle
T_{1}\hat{\tau}_{m}=T_{1}A_{1}\pi^{(1)}_{m}=A_{1}T_{0}\pi^{(1)}_{m}=\lambda_{m}\hat{\tau}_{m};$

i.e., $\hat{\tau}_{m}$ is a factorization eigenfunction of $T_{1}$ at
$\lambda_{m}$. By Proposition 4.6, we have thus

$\displaystyle B^{(\alpha)}_{\tau_{m},\pi}A_{\pi,\tau_{m}}$
$\displaystyle=T^{(\alpha)}_{\tau_{m}}-\lambda$ $\displaystyle
A_{\pi,\tau_{m}}B^{(\alpha)}_{\tau_{m},\pi}$
$\displaystyle=T^{(\alpha+1)}_{\pi}-2\alpha-1-\lambda.$

We conclude therefore that $T_{2}A_{2}=A_{2}T_{1}$, as was to be shown. ∎

######  Proposition 4.10.

The polynomials $\pi_{m,i}(z;t)$ as defined in (57) are eigenfunctions of
$T^{(\alpha)}_{\tau_{m}}$.

###### Proof.

Combining the intertwining relations in the preceding proof, we have the
second-order intertwining relation

(62) $A_{21}T_{0}=T_{2}A_{21},\quad\text{where }A_{21}:=A_{2}A_{1}.$

Our first claim is that

(63) $A_{21}\pi_{i}=(\lambda_{i}-\lambda_{m})\pi_{m,i},\quad i\neq m.$

Let $i\in\mathbb{N}_{0}$ be given. To establish the previous equation, we
consider $A_{1}$ and $A_{2}$ in terms of their Wronskian formulations:

$\displaystyle A_{1}y$
$\displaystyle=A_{\tau\pi_{m}}y=\tau^{-1}\operatorname{\operatorname{Wr}}[\pi_{m},y],$
$\displaystyle A_{2}y$
$\displaystyle=B^{(\alpha)}_{\pi_{m}\tau_{m}}y=\hat{\pi}_{m}^{-1}\operatorname{\operatorname{Wr}}[\hat{\tau}_{m},y],$
where $\displaystyle\hat{\pi}_{m}(z)$
$\displaystyle=(1-z^{2})^{-\alpha-\frac{3}{2}}\pi_{m}(z).$

Lagrange’s identity (45) implies that

$\displaystyle\operatorname{\operatorname{Wr}}[\pi_{m},\pi_{i}]=(\lambda_{i}-\lambda_{m})\mu\,\rho_{mi}.$

Hence,

$\displaystyle
A_{1}\pi_{i}=\tau^{-1}\operatorname{\operatorname{Wr}}[\pi_{m},\pi_{i}]=(\lambda_{i}-\lambda_{m})\hat{\tau}\rho_{mi}.$

We define $B_{1}:=B^{(\alpha)}_{\pi_{m},\tau}$ and note that by Proposition
4.5 we have the factorization $T_{0}=B_{1}A_{1}+\lambda_{m}$. By the linearity
of the Wronskian, we also have

$\displaystyle A_{2}y$
$\displaystyle=\hat{\pi}_{m}^{-1}\operatorname{\operatorname{Wr}}[\hat{\tau}_{m},y]=\hat{\pi}_{m}^{-1}\operatorname{\operatorname{Wr}}[\hat{\tau}(1+t\rho_{mm}),y]$
$\displaystyle=\hat{\pi}_{m}^{-1}\operatorname{\operatorname{Wr}}[\hat{\tau},y]+\hat{\pi}_{m}^{-1}\operatorname{\operatorname{Wr}}[t\hat{\tau}\rho_{mm},y]$
$\displaystyle=B_{1}y+t\hat{\pi}_{m}^{-1}\operatorname{\operatorname{Wr}}[\hat{\tau}\rho_{mm},y]$

Finally, we have that

$\displaystyle A_{21}\pi_{i}$
$\displaystyle=B_{1}A_{1}\pi_{i}+t(\lambda_{i}-\lambda_{m})\hat{\pi}_{m}^{-1}\operatorname{\operatorname{Wr}}[\hat{\tau}\rho_{mm},\hat{\tau}\rho_{mi}]$
$\displaystyle=(T_{0}-\lambda_{m})\pi_{i}+(\lambda_{i}-\lambda_{m})t\hat{\tau}^{2}\hat{\pi}_{m}^{-1}\operatorname{\operatorname{Wr}}[\rho_{mm},\rho_{mi}]$
$\displaystyle=(\lambda_{i}-\lambda_{m})(\pi_{i}+t\hat{\tau}^{2}\hat{\pi}_{m}^{-1}(\rho_{mm}\pi_{m}\pi_{i}{W^{(\alpha)}_{\tau}}-\rho_{mi}\pi_{m}^{2}{W^{(\alpha)}_{\tau}})$
$\displaystyle=(\lambda_{i}-\lambda_{m})(\pi_{i}+t(\rho_{mm}\pi_{i}-\rho_{mi}\pi_{{m}})).$

This establishes (63).

Next suppose that $i\neq m$. Then, by (62), it follows that

(64) $\displaystyle
T_{2}\pi_{m;i}=(\lambda_{i}-\lambda_{m})^{-1}T_{2}A_{21}\pi_{i}=(\lambda_{i}-\lambda_{m})^{-1}A_{21}T_{0}\pi_{i}\lambda_{i}\pi_{m;i}.$

Finally, observe that $\pi_{m,m}=\pi_{m}$ and recall that

$T_{2}=B^{(\alpha)}_{\pi_{m},\tau_{m}}A_{\tau_{m},\pi_{m}}+\lambda_{m}.$

Hence, $T_{2}\pi_{m}=\lambda_{m}\pi_{m}$, as was to be shown. ∎

The last proposition of this section shows that the CDT of an exceptional
Gegenbauer family falls into the same class provided we impose suitable bounds
on the introduced parameter $t$. The form of the norming constants of the new
eigenpolynomials relative to the weight $W^{(\alpha)}_{\tau_{m}}$ follow as a
direct corollary to this proposition. We denote the norm of $\pi_{i}$ by
$\nu_{i}$, and similarly the norm of $\pi_{m;i}$ is $\nu_{m;i}$. Explicitly,
we have

(65) $\displaystyle\nu_{i}$
$\displaystyle:=\int_{-1}^{1}\pi_{i}(s)^{2}W^{(\alpha)}_{\tau}(s)ds=\rho_{ii}(1),\quad
i\in\mathbb{N}_{0};$ (66) $\displaystyle\nu_{m;i}$
$\displaystyle:=\int_{-1}^{1}\pi_{m;i}(s)^{2}W^{(\alpha)}_{\tau_{m}}(s)ds=\rho_{m;ii}(1),\quad
m,i\in\mathbb{N}_{0};$ where (67) $\displaystyle\rho_{m;ij}(z)$
$\displaystyle:=\int_{-1}^{z}\pi_{m;i}(u;t)\pi_{m;j}(u;t)W^{(\alpha)}_{\tau_{m}}(u)du,\quad
m,i,j\in\mathbb{N}_{0}.$

The following proposition exhibits a recursive formulation for the functions
$\rho_{m;ij}$. As a consequence, if the $\rho_{ij}(z)$ are rational, then so
are $\rho_{m;ij}(z)$, i.e. rationality is preserved by a CDT.

######  Proposition 4.11.

Let $\rho_{ij},\tau_{m},\pi_{m;i},\rho_{m;ij}$ be defined by (55)–(57) and
(67). Then,

(68)
$\displaystyle\rho_{m;ij}(z,t)=\rho_{ij}(z)-\frac{t\rho_{mi}(z)\rho_{mj}(z)}{1+t\rho_{mm}(z)},\quad
m,i,j\in\mathbb{N}_{0}.$

###### Proof.

Observe that

$\displaystyle\left(\rho_{ij}-\frac{t\rho_{mi}\rho_{mj}}{1+t\rho_{mm}}\right)^{\prime}=\rho_{ij}^{\prime}-\frac{t\rho_{mi}^{\prime}\rho_{mj}}{1+t\rho_{mm}}-\frac{t\rho_{mi}\rho_{mj}^{\prime}}{1+t\rho_{mm}}+\frac{t^{2}\rho_{mi}\rho_{mj}}{(1+t\rho_{mm})^{2}}\rho_{mm}^{\prime}$
$\displaystyle\quad=\Big{(}(1+t\rho_{mm})^{2}\pi_{i}\pi_{j}-t\pi_{m}(1+t\rho_{mm})(\pi_{i}\rho_{mj}+\pi_{j}\rho_{mi})+t^{2}\rho_{mi}\rho_{mj}\pi_{m}^{2}\Big{)}\frac{W^{(\alpha)}_{\tau}}{(1+t\rho_{mm})^{2}}$
$\displaystyle\quad=\Big{(}(1+t\rho_{mm})\pi_{i}-t\rho_{mi}\pi_{m}\Big{)}\Big{(}(1+t\rho_{mm})\pi_{j}-t\rho_{mj}\pi_{m}\Big{)}W^{(\alpha)}_{\tau_{m}}$
$\displaystyle\quad=\pi_{m;i}\pi_{m;j}W^{(\alpha)}_{\tau_{m}}.$

The desired result follows then by integration since $\rho_{ij}(-1)=0$ by
definition, and $\rho_{ij}(z)$ is rational by assumption. ∎

In order to demonstrate that the $\pi_{m;i}(z)$ are exceptional Gegenbauer
polynomials in the sense of Definition 4.2, we must first establish the
positivity of $\tau_{m}$ on $[-1,1]$ conditioned on the positivity of $\tau$.

######  Proposition 4.12.

Suppose that $\tau(z)>0$ for all $z\in I=[-1,1]$. Then, $\tau_{m}$ is positive
on $I$ if and only if $1+t\nu_{m}>0$.

###### Proof.

By (56), $\tau_{m}$ is positive on $I$ if and only if the same is true for
$1+t\rho_{mm}(z)$. We know that $\rho_{mm}(z)$ is an increasing function,
since

$\displaystyle\rho_{mm}^{\prime}=\pi_{m}^{2}\frac{(1-z^{2})^{\alpha-1/2}}{\tau^{2}}>0.$

Additionally, we notice that $\rho_{mm}$ is differentiable, and hence
continuous, on the interval $I$. Fix $t$ and define $g(z):=1+t\rho_{mm}(z)$,
so that $g^{\prime}(z)=t\rho_{mm}^{\prime}(z)$. If $t\geq 0$, then $g$ is
positive on $I$, since $\rho_{mm}$ is an increasing function and $g(-1)=1$.
However, if $t<0$, then $g$ is a decreasing function on $I$. Importantly, $g$
cannot have any local extrema on the interval. Hence, if $g(1)=1+t\nu_{m}>0$,
then $g$ must be positive on all of $I$. Hence, $1+t\rho_{mm}(z)$ is positive
on $z\in I$ if and only if $1+t\nu_{m}>0$. ∎

######  Proposition 4.13.

Let $\pi_{m;i}(z,t)$ be defined as in (57), and assume that $1+t\nu_{m}>0$.
Then, the norms (65) and (66) are related by

(69) $\nu_{m;i}^{-1}=\nu_{m}^{-1}+\delta_{im}t.$

###### Proof.

By Proposition 4.11, we have that

$\displaystyle\nu_{m;i}=\nu_{i}-\frac{t\rho_{mi}(1)^{2}}{1+t\rho_{mm}(1)}=\nu_{i},\quad
i\neq m,$

Suppose that $i\neq m$. Then, $\pi_{i}$ and $\pi_{m}$ for $i\neq m$ are
orthogonal relative to $W^{(\alpha)}_{\tau}$. Hence, $\rho_{mi}(1)=0$ and
$\nu_{m;m}=\nu_{m}$. By definition, $\nu_{m}=\rho_{mm}(1)$. Hence, if $i=m$,
we have

$\nu_{m;m}=\nu_{m}-\frac{t\nu_{m}^{2}}{1+t\nu_{m}}=\frac{\nu_{m}}{1+t\nu_{m}}=(t+\nu_{m}^{-1})^{-1}.$

∎

######  Remark 4.14.

As a consequence of (66), we can recover the deformation parameter as the
difference of the following norm reciprocals:

$t=\nu_{m;m}^{-1}-\nu_{m}^{-1}.$

Consequently,

$\displaystyle\nu_{m;m}^{-1}=\nu_{m}^{-1}(1+t\nu_{m}).$

Hence, by Proposition 4.12, the positivity of $\tau_{m}$ is equivalent to the
condition that $\nu_{m;m}>0$.

######  Proposition 4.15.

Let $\alpha\in\mathbb{N}_{0}+1/2$ and suppose that
$\\{\pi_{i}(z)\\}_{i\in\mathbb{N}_{0}}$ are exceptional Gegenbauer polynomials
with respect to the weight $W^{(\alpha)}_{\tau}(z)$. Let $m\in\mathbb{N}_{0}$
and suppose that $1+t\nu_{m}>0$. Then, the family
$\\{\pi_{m;i}(z)\\}_{i\in\mathbb{N}_{0}}$ are also exceptional Gegenbauer
polynomials with respect to the weight $W^{(\alpha)}_{\tau_{m}}(z)$.

###### Proof.

Proposition 4.12 establishes condition (a) of Definition 4.2. Proposition 4.9
establishes (b). It remains to prove that the completeness condition (c) also
holds. Set

(70) $\displaystyle\tilde{\pi}_{i}(z)$
$\displaystyle:=\left(W^{(\alpha)}_{\tau}(z)\right)^{\frac{1}{2}}\pi_{i}(z)=(1-z^{2})^{\frac{\alpha}{2}-\frac{1}{4}}\frac{\pi(z)}{\tau(z)},$
$\displaystyle\tilde{\pi}_{m;i}(z)$
$\displaystyle:=\left(W^{(\alpha)}_{\tau_{m}}(z)\right)^{\frac{1}{2}}\pi_{m;i}(z)=(1-z^{2})^{\frac{\alpha}{2}-\frac{1}{4}}\frac{\pi_{m;i}(z)}{\tau_{m}(z)}.$

By assumption, the eigenpolynomials $\\{\pi_{i}(z)\\}_{i\in\mathbb{N}_{0}}$
form a complete basis of $\mathrm{L}^{2}(I,W^{(\alpha)}_{\tau}(z)dz)$.
Equivalently, $\\{\tilde{\pi}_{i}(z)\\}_{i\in\mathbb{N}_{0}}$ are complete in
$\mathrm{L}^{2}(I,dz)$. We seek to show that
$\\{\tilde{\pi}_{m;i}(z)\\}_{i\in\mathbb{N}_{0}}$ are complete in
$\mathrm{L}^{2}(I,dz)$ also.

Following an argument adapted from the appendix of [1], we observe that the
completeness of $\\{\tilde{\pi}_{i}\\}_{i\in\mathbb{N}_{0}}$ in
$\mathrm{L}^{2}(I,dz)$ is equivalent to

(71)
$\sum_{i\in\mathbb{N}_{0}}\nu_{i}^{-1}\tilde{\pi}_{i}(z)\tilde{\pi}_{i}(w)=\delta(z-w),$

where the equality must be understood in distributional sense on $I\times I$.
Let $f$ be piece-wise continuous function in $I$ and set

(72) $[f]_{i}=\nu_{i}^{-1}\int_{I}\tilde{\pi}_{i}(u)f(u)du.$

Relation (71) then entails

(73)
$\sum_{i\in\mathbb{N}_{0}}[f]_{i}\,\tilde{\pi}_{i}(z)=f(z),\quad\text{a.e.
}z\in I.$

Moreover, if $g$ is another piece-wise continuous in $I$ function, then

(74) $\int_{I}f(u)g(u)du=\sum_{i\in\mathbb{N}_{0}}\nu_{i}[f]_{i}[g]_{i}.$

For a given $w\in I$, and a function $f(z),\;z\in I$, define the truncation
operator

$(\mathcal{T}_{w}f)(z):=\theta(w-z)f(z),\quad z\in I,$

where

(75) $\theta(u):=\begin{cases}1&\text{ if }u>0,\\\ \frac{1}{2}&\text{ if
}u=0,\\\ 0&\text{ if }u<0;\end{cases}$

denotes the Heaviside step function. We may now rewrite (55) as

$\rho_{ij}(w)=\int_{I}(\mathcal{T}_{w}\tilde{\pi}_{j})(u)\,\tilde{\pi}_{i}(u)\,du=\nu_{i}[\mathcal{T}_{w}\tilde{\pi}_{j}]_{i},\quad
i,j\in\mathbb{N}_{0},\;w\in I.$

Applying (73) then gives

(76)
$\theta(w-z)\tilde{\pi}_{j}(z)=(\mathcal{T}_{w}\tilde{\pi}_{j})(z)=\sum_{i\in\mathbb{N}_{0}}\nu_{i}^{-1}\rho_{ij}(w)\tilde{\pi}_{i}(z),\quad
z,w\in I.$

Moreover, (74) implies that

$\displaystyle\sum_{i\in\mathbb{N}_{0}}\nu_{i}^{-1}\rho_{ij}(z)\rho_{ij}(w)$
$\displaystyle=\int_{I}(\mathcal{T}_{w}\tilde{\pi}_{j})(u)(\mathcal{T}_{z}\tilde{\pi}_{j})(u)du,\quad
z,w\in I,\;j\in\mathbb{N}_{0}.$

Observe that

$\theta(z)\theta(w)=\theta(\min(z,w)),\quad z,w\in I.$

Hence,

$\int_{I}(\mathcal{T}_{w}\tilde{\pi}_{j})(u)(\mathcal{T}_{z}\tilde{\pi}_{j})(u)du=\int_{I}\theta(w-u)\theta(z-u)\tilde{\pi}_{j}(u)\tilde{\pi}_{j}(u)\,du=\rho_{jj}(\min(z,w)).$

Therefore,

(77)
$\sum_{i\in\mathbb{N}_{0}}\nu_{i}^{-1}\rho_{ij}(z)\rho_{ij}(w)=\theta(w-z)\rho_{jj}(z)+\theta(z-w)\rho_{jj}(w).$

By (56), (57) and (70),

(78) $\displaystyle 1+t\rho_{mm}$ $\displaystyle=\frac{\tau_{m}}{\tau};$ (79)
$\displaystyle\tilde{\pi}_{m;i}$
$\displaystyle=(1-z^{2})^{\frac{\alpha}{2}-\frac{1}{4}}\frac{1}{\tau_{m}}\left(\frac{\tau_{m}}{\tau}\pi_{i}-t\rho_{im}\pi_{m}\right)=\tilde{\pi}_{i}-t\rho_{im}\tilde{\pi}_{m;m};$
(80) $\displaystyle\tilde{\pi}_{m}$
$\displaystyle=(1+t\rho_{mm})\tilde{\pi}_{m;m}.$

By (66), (79), (71), (76), (77) and (80), we have

$\displaystyle\sum_{i\in\mathbb{N}_{0}}$
$\displaystyle\nu_{m;i}^{-1}\tilde{\pi}_{m;i}(z)\tilde{\pi}_{m;i}(w)$
$\displaystyle=t\tilde{\pi}_{m;m}(z)\tilde{\pi}_{m;m}(w)+\sum_{i}\nu_{i}^{-1}\tilde{\pi}_{m;i}(z)\tilde{\pi}_{m;i}(w)$
$\displaystyle=t\tilde{\pi}_{m;m}(z)\tilde{\pi}_{m;m}(w)+\sum_{i}\nu_{i}^{-1}\left(\tilde{\pi}_{i}(z)-t\rho_{im}(z)\tilde{\pi}_{m;m}(z)\right)\left(\tilde{\pi}_{i}(w)-t\rho_{im}(w)\tilde{\pi}_{m;m}(w)\right)$
$\displaystyle=\sum_{i}\nu_{i}^{-1}\tilde{\pi}_{i}(z)\tilde{\pi}_{i}(w)+t\tilde{\pi}_{m;m}(z)\tilde{\pi}_{m;m}(w)\left(1+t\sum_{i}\nu_{i}^{-1}\rho_{im}(z)\rho_{im}(w)\right)$
$\displaystyle\quad-t\,\tilde{\pi}_{m;m}(z)\sum_{i}\nu_{i}^{-1}\rho_{im}(z)\tilde{\pi}_{i}(w)-t\,\tilde{\pi}_{m;m}(w)\sum_{i}\nu_{i}^{-1}\rho_{im}(w)\tilde{\pi}_{i}(z)$
$\displaystyle=\delta(z-w)+t\tilde{\pi}_{m;m}(z)\tilde{\pi}_{m;m}(w)\left(1+\theta(w-z)t\rho_{mm}(z)+\theta(z-w)t\rho_{mm}(w)\right)$
$\displaystyle\quad-t\,\theta(z-w)\tilde{\pi}_{m;m}(z)\tilde{\pi}_{m}(w)-t\,\theta(w-z)\tilde{\pi}_{m}(z)\tilde{\pi}_{m;m}(w)$
$\displaystyle=\delta(z-w)+t\tilde{\pi}_{m;m}(z)\tilde{\pi}_{m;m}(w)\Big{(}1+\theta(w-z)t\rho_{mm}(z)+\theta(z-w)t\rho_{mm}(w)$
$\displaystyle\qquad\qquad-\theta(z-w)(1+t\rho_{mm}(w))-\theta(w-z)(1+t\rho_{mm}(z))\Big{)}$
$\displaystyle=\delta(z-w)+t\tilde{\pi}_{m;m}(z)\tilde{\pi}_{m;m}(w)\Big{(}1-\theta(w-z)-\theta(z-w)\Big{)}$
$\displaystyle=\delta(z-w).$

We conclude that the set $\\{\tilde{\pi}_{m;i}\\}_{i\in\mathbb{N}_{0}}$ is
complete in $\mathrm{L}^{2}(I,dz)$, and by virtue of the previously stated
equivalence, the set $\\{\pi_{m;i}\\}_{i\in\mathbb{N}_{0}}$ is complete in
$\mathrm{L}^{2}(I,W^{(\alpha)}_{\tau_{m}}(z)dz)$. ∎

Note that in Proposition 4.15 and in fact, all throughout the current Section,
we have assumed that $\pi_{m,i}$ are polynomials, which is not guaranteed by
their defining expressions (55)-(57) and (39), even if $\pi$ are polynomials
and $\alpha\in\mathbb{N}_{0}+1/2$ (but they are certainly not polynomials if
$\alpha$ is not half-integer). In the following Section we will show that when
these objects are connected to the classical Gegenbauer polynomials by a chain
of confluent Darboux transformations, the assumptions on polynomiality are
guaranteed at each step of the chain.

##  5\. Exceptional Gegenbauer polynomials of the second kind

In this Section we provide explicit formulas for the construction of
exceptional Gegenbauer polynomials of the second kind and their associated
operators in terms of a matrix whose entries involve classical Gegenbauer
polynomials. Throughout this section, we also assume that
$\alpha\in\mathbb{N}_{0}+\frac{1}{2}$ is a positive half-integer. Note that,
with this assumption in place, the weight $W^{(\alpha)}(z)$ in (47) becomes a
polynomial.

We next define several objects that allow us to describe the exceptional
Gegenbauer polynomials and operators below. Set

(81)
$\rho^{(\alpha)}_{ij}(z):=\int_{-1}^{z}C^{(\alpha)}_{i}(u)C^{(\alpha)}_{j}(u)W^{(\alpha)}\\!(u)\,du,\quad
i,j\in\mathbb{N}_{0}$

and observe that the above functions are polynomials precisely because
$\alpha$ is a positive half-integer. Given an $n$–tuple
${\boldsymbol{m}}\in\mathbb{N}_{0}^{n}$ and the associated
$\boldsymbol{t}_{\boldsymbol{m}}\in\mathbb{R}^{n}$, we then define
${\mathcal{R}}^{(\alpha)}_{{\boldsymbol{m}}}={\mathcal{R}}^{(\alpha)}_{{\boldsymbol{m}}}(z;\boldsymbol{t}_{\boldsymbol{m}})$
as the $n\times n$ matrix with polynomial entries given by

(82)
$\displaystyle[{\mathcal{R}}^{(\alpha)}_{{\boldsymbol{m}}}]_{k\ell}=\delta_{k\ell}+t_{m_{\ell}}\rho^{(\alpha)}_{m_{k}m_{\ell}}(z),\quad
k,\ell\in\\{1,\dots,n\\},$

We denote its determinant by

(83)
$\displaystyle\tau^{(\alpha)}_{{\boldsymbol{m}}}:=\det{\mathcal{R}}^{(\alpha)}_{\boldsymbol{m}}.$

Next, define the $n$–tuple of polynomials

(84)
$\displaystyle\left({\boldsymbol{Q}}^{(\alpha)}_{{\boldsymbol{m}}}{}\right)^{T}:=\tau^{(\alpha)}_{{\boldsymbol{m}}}\left({\mathcal{R}}^{(\alpha)}_{{\boldsymbol{m}}}\right)^{-1}\left(C^{(\alpha)}_{m_{1}},\ldots,C^{(\alpha)}_{m_{n}}\right)^{T},$

We are now ready to define the fundamental objects of this Section.

######  Definition 5.1.

Let ${\boldsymbol{m}}=(m_{1},\dots,m_{n})\in\mathbb{N}_{0}^{n}$ be a tuple of
distinct integers and ${\boldsymbol{Q}}^{(\alpha)}_{{\boldsymbol{m}}}$ the
$n$-tuple of polynomials defined by (81)-(84). We define the _exceptional
Gegenbauer polynomials of the second kind_ associated to ${\boldsymbol{m}}$ as

(85) $\displaystyle
C^{(\alpha)}_{{\boldsymbol{m}};i}:=\big{[}{\boldsymbol{Q}}^{(\alpha)}_{({\boldsymbol{m}},i)}\big{]}_{n+1},\quad
i\in\mathbb{N}_{0}.$

######  Remark 5.2.

Note that, by construction,
$\tau^{(\alpha)}_{\boldsymbol{m}}=\tau^{(\alpha)}_{{\boldsymbol{m}}}(z;\boldsymbol{t}_{\boldsymbol{m}})$
is invariant with respect to permutations of the indices
${\boldsymbol{m}}=(m_{1},\ldots,m_{n})$ and that
${\boldsymbol{Q}}^{(\alpha)}_{{\boldsymbol{m}}}={\boldsymbol{Q}}^{(\alpha)}_{{\boldsymbol{m}}}(z;\boldsymbol{t}_{{\boldsymbol{m}}})$
is equivariant with respect to such permutations. In addition,
$C^{(\alpha)}_{{\boldsymbol{m}};i}=C^{(\alpha)}_{{\boldsymbol{m}};i}(z;\boldsymbol{t}_{\boldsymbol{m}})$
is symmetric in ${\boldsymbol{m}}$ and does not depend on $t_{i}$ since
$\tau^{(\alpha)}_{({\boldsymbol{m}},i)}\big{[}\left({\mathcal{R}}^{(\alpha)}_{({\boldsymbol{m}},i)}\right)^{-1}\big{]}_{n+1,j}$
correspond to the minors of the last column of
${\mathcal{R}}^{(\alpha)}_{({\boldsymbol{m}},i)}$, the only column where
$t_{i}$ appears.

The main result of this section states that the polynomials
$\big{\\{}C^{(\alpha)}_{{\boldsymbol{m}};i}(z;\boldsymbol{t}_{\boldsymbol{m}})\big{\\}}_{i\in\mathbb{N}_{0}}$
defined above are indeed exceptional Gegenbauer polynomials (in the sense of
Definition 4.2), provided the real parameters
$\boldsymbol{t}_{\boldsymbol{m}}$ satisfy certain constraints to ensure that
$\tau^{(\alpha)}_{\boldsymbol{m}}(z;t_{\boldsymbol{m}})$ is positive on
$z\in[-1,1]$. We first state that the polynomial
$\tau_{{\boldsymbol{m}}}^{(\alpha)}$ inserted in (38) is an exceptional
Gegenbauer operator in the sense of Definition 4.1.

######  Theorem 5.1.

Let $\alpha\in\mathbb{N}_{0}+\frac{1}{2}$ and
${\boldsymbol{m}}\in\mathbb{N}_{0}^{n}$. Consider the $n$-parameter family of
operators
$T^{(\alpha)}_{{\boldsymbol{m}}}:=T^{(\alpha)}_{\tau^{(\alpha)}_{{\boldsymbol{m}}}}$
given by (38) and (83). For each value of the parameters
$\boldsymbol{t}_{{\boldsymbol{m}}}$, this operator is an exceptional
Gegenbauer operator that satisfies

(86)
$T^{(\alpha)}_{{\boldsymbol{m}}}C^{(\alpha)}_{{\boldsymbol{m}};i}=\lambda_{i}C^{(\alpha)}_{{\boldsymbol{m}};i},\quad
i\in\mathbb{N}_{0},$

with $\lambda_{i}=-i(2\alpha+i)$.

The following theorem provides necessary and sufficient conditions for the
polynomials $\\{C^{(\alpha)}_{{\boldsymbol{m}};i}\\}_{i\in\mathbb{N}_{0}}$ to
be a family of exceptional Gegenbauer polynomials according to Definition 4.2.
If these conditions hold the family
$\\{C^{(\alpha)}_{{\boldsymbol{m}};i}(z;\boldsymbol{t}_{\boldsymbol{m}})\\}_{i\in\mathbb{N}_{0}}$
is orthogonal and complete, like their classical counterparts.

######  Theorem 5.2.

Let $\alpha\in\mathbb{N}_{0}+\frac{1}{2}$ and
${\boldsymbol{m}}\in\mathbb{N}_{0}^{n}$ with $m_{1},\ldots,m_{n}$ distinct.
Then the polynomial
$\tau^{(\alpha)}_{\boldsymbol{m}}(z;\boldsymbol{t}_{\boldsymbol{m}})$ in (83)
has no zeros on $z\in[-1,1]$ if and only if

(87) $t_{m_{j}}>-\left(\nu^{(\alpha)}_{m_{j}}\right)^{-1},\quad j=1,\ldots,n,$

with $\nu^{(\alpha)}_{m_{j}}$ as in (48). If the above conditions hold, then
$\big{\\{}C^{(\alpha)}_{{\boldsymbol{m}};i}(z;\boldsymbol{t}_{\boldsymbol{m}})\big{\\}}_{i\in\mathbb{N}_{0}}$
are exceptional Gegenbauer polynomials with weight $W^{(\alpha)}_{\tau_{m}}$
and norms given by

(88)
$\int_{I}\left(C^{(\alpha)}_{{\boldsymbol{m}};i}(u)\right)^{2}W^{(\alpha)}_{\tau_{m}}(u)du\
=\frac{\nu^{(\alpha)}_{i}}{1+\delta_{i,{\boldsymbol{m}}}t_{i}\nu^{(\alpha)}_{i}}$

where

(89) $\delta_{i,{\boldsymbol{m}}}:=\begin{cases}1&\text{ if
}i\in\\{m_{1},\ldots,m_{n}\\};\\\ 0&\text{ otherwise.}\end{cases}$

As mentioned above, the degree of the $i$-th exceptional Gegenbauer polynomial
$C^{(\alpha)}_{{\boldsymbol{m}};i}$ is not necessarily $i$. The next
proposition provides this result. It is also worth noting that, as opposed to
the exceptional families of the first kind, the degree sequence of the
exceptional Gegenbauer polynomials is not an increasing sequence, which is
further evidence of the different construction.

######  Proposition 5.3.

Let $\alpha\in\mathbb{N}_{0}+\frac{1}{2}$ and
${\boldsymbol{m}}\in\mathbb{N}_{0}^{n}$ with $m_{1},\ldots,m_{n}$ distinct.
Let $\tau^{(\alpha)}_{\boldsymbol{m}},C^{(\alpha)}_{{\boldsymbol{m}};i}$ be as
defined in (83) and (85). Then,

(90)
$\displaystyle\deg_{z}\tau^{(\alpha)}_{{\boldsymbol{m}}}=2(m_{1}+\ldots+m_{n}+\alpha
n),$ (91)
$\displaystyle\deg_{z}C^{(\alpha)}_{{\boldsymbol{m}};i}=2(m_{1}+\ldots+m_{n}+\alpha
n)+i-2\delta_{i,{\boldsymbol{m}}}(i+\alpha),\quad i\in\mathbb{N}_{0}.$

Moreover,

(92)
$C^{(\alpha)}_{{\boldsymbol{m}};m_{k}}=C^{(\alpha)}_{m_{1},\ldots,\widehat{m_{k}},\ldots,m_{n};m_{k}},\quad
k=1,\ldots,n,$

where the hat symbol denotes the omission of the $k^{\text{th}}$ entry of
${\boldsymbol{m}}$.

######  Remark 5.4.

From Proposition 5.3, we see that the codimension (number of missing degrees)
of the exceptional Gegenbauer family indexed by
${\boldsymbol{m}}=(m_{1},\dots,m_{n})$ is $2(m_{1}+\cdots+m_{n}+\alpha n)$. As
it happens for all exceptional polynomials [12], this coincides with the
degree of $\tau^{(\alpha)}_{\boldsymbol{m}}$.

######  Remark 5.5.

Below we will show that the $n$-parameter family
$\big{\\{}C^{(\alpha)}_{{\boldsymbol{m}};i}(z;t_{\boldsymbol{m}})\big{\\}}_{i\in\mathbb{N}_{0}}$
is the result of applying $n$ confluent Darboux transformations to the
classical family $\big{\\{}C^{(\alpha)}_{i}(z)\big{\\}}_{i\in\mathbb{N}_{0}}$.
Identity (88) then tells us that a single CDT leaves invariant all but one of
the norming constants. By contrast, identity (92) tells us that after a single
CDT there is exactly one polynomial that remains the same but whose norm
undergoes a change. It is this phenomenon that accounts for the Kronecker
delta term in (91).

######  Remark 5.6.

The degree formulas (90), (91) allow us to relate the eigenvalue formula (41)
with the value $\lambda_{i}=-i(2\alpha+i)$ given in Theorem 5.1. Indeed if
$i\notin\\{m_{1},\ldots,m_{n}\\}$ then $\deg
C^{(\alpha)}_{{\boldsymbol{m}},i}-\deg\tau^{(\alpha)}_{\boldsymbol{m}}=i$ in
full agreement with (41). By contrast, if $i=m_{k}$ for some
$k\in\\{1,\ldots,n\\}$ then

$d_{i}:=\deg
C^{(\alpha)}_{{\boldsymbol{m}},i}-\deg\tau^{(\alpha)}_{\boldsymbol{m}}=-i-2\alpha$

and therefore

$d_{i}(2\alpha+d_{i})=(i+2\alpha)i.$

Hence, (41) is correct in this case also.

In Theorem 5.2 and Proposition 4.3 we have considered the case when
${\boldsymbol{m}}=(m_{1},\dots,m_{n})$ contains distinct indices. Let us now
show that this choice entails no loss of generality. Indeed, the repeated
application of a confluent Darboux transformation at the same eigenvalue only
serves to modify the deformation parameter.

######  Proposition 5.7.

Let $\alpha\in\mathbb{N}_{0}+\frac{1}{2}$,
${\boldsymbol{m}}\in\mathbb{N}_{0}^{n}$ and let
$\tau^{(\alpha)}_{\boldsymbol{m}},C^{(\alpha)}_{{\boldsymbol{m}};i}$ be as
defined in (83) and (85). Then, for any $j\in\mathbb{N}_{0}$, we have

$\displaystyle\tau^{(\alpha)}_{({\boldsymbol{m}},j,j)}(z;{\color[rgb]{0,0,1}(}\boldsymbol{t}_{\boldsymbol{m}},t_{j},t_{j}^{\prime}{\color[rgb]{0,0,1})})=\tau^{(\alpha)}_{({\boldsymbol{m}},j)}(z;{\color[rgb]{0,0,1}(}\boldsymbol{t}_{\boldsymbol{m}},t_{j}+t^{\prime}_{j}{\color[rgb]{0,0,1})}),$
$\displaystyle
C^{(\alpha)}_{({\boldsymbol{m}},j,j);i}(z;(\boldsymbol{t}_{\boldsymbol{m}},t_{j},t_{j}^{\prime}))=C^{(\alpha)}_{({\boldsymbol{m}},j);i}(z;({\color[rgb]{0,0,1}(}\boldsymbol{t}_{\boldsymbol{m}},t_{j}+t^{\prime}_{j})).$

###  5.1. Proof of theorems

In this section we provide proofs for all of the Theorems and Propositions in
this Section.

  1. 1.

We define polynomials $\tilde{\tau}_{j}$, $\tilde{\pi}_{0;i}$ and rational
functions $\tilde{\rho}_{j;i_{1}i_{2}}$ recursively, starting the recursion at
the objects corresponding to the classical Gegenbauer Sturm–Liouville problem.

  2. 2.

We show that these recursion formulas describe a multi-step confluent Darboux
transformation.

  3. 3.

We show in Proposition 5.9 that the recursively defined objects coincide with
those obtained via the matrix-based definitions (81)-(85).

  4. 4.

Since the objects defined by the matrix formulas are polynomial by
construction, we can dispense with the rationality and polynomiality
assumptions made at the beginning of Section 4.3.

  5. 5.

Propositions 4.9–4.15 then ensure that at each step of the recursion we have
an exceptional Gegenbauer Sturm–Liouville problem, provided the parameters are
chosen in the right range.

Fix
${\boldsymbol{m}}={\boldsymbol{m}}_{n}=(m_{1},\dots,m_{n})\in\mathbb{N}_{0}^{n}$
and a positive half-integer $\alpha\in\mathbb{N}_{0}+\frac{1}{2}$. For
$j=0,1,\ldots,n$ let ${\boldsymbol{m}}_{j}\in\mathbb{N}_{0}^{n}$ denote the
initial segment of ${\boldsymbol{m}}$; i.e.,
${\boldsymbol{m}}_{j}=(m_{1},\dots,m_{j})$. Note that, throughout this
section, we are going to omit the explicit dependence on $z$ and
$\boldsymbol{t}_{\boldsymbol{m}}$ of the objects, which must be understood
from the dependence on ${\boldsymbol{m}}$, i.e., we will write
${\mathcal{R}}^{(\alpha)}_{\boldsymbol{m}}$ instead of
${\mathcal{R}}^{(\alpha)}_{\boldsymbol{m}}(z;\boldsymbol{t}_{\boldsymbol{m}})$.
In order to simplify the notation, we will sometimes make the dependence of
the various objects on $\alpha$ implicit rather than explicit.

We start the recursion at $j=0$ by setting

$\displaystyle\tilde{\rho}_{0;i_{1}i_{2}}:=\rho^{(\alpha)}_{i_{1}i_{2}},\quad\tilde{\tau}_{0}:=1,\quad\tilde{\pi}_{0;i}:=C^{(\alpha)}_{i},$

where $\rho^{(\alpha)}_{i_{1}i_{2}}(z)$ is given by (81) and
$C^{(\alpha)}_{i}(z)$ are the classical Gegenbauer polynomials (46). For
$j=1,\dots,n$ we then define

(93) $\displaystyle\tilde{\tau}_{j}$
$\displaystyle=\left(1+t_{m_{j}}\tilde{\rho}_{j-1;m_{j}m_{j}}\right)\tilde{\tau}_{j-1},$
(94) $\displaystyle\tilde{\pi}_{j;i}$
$\displaystyle=\left(1+t_{m_{j}}\tilde{\rho}_{j-1;m_{j}m_{j}}\right)\tilde{\pi}_{j-1;i}-t_{m_{j}}\tilde{\rho}_{j-1;im_{j}}\tilde{\pi}_{j-1;m_{j}},\quad
i\in\mathbb{N}_{0};$ (95) $\displaystyle\tilde{\rho}_{j;i_{1}i_{2}}$
$\displaystyle=\tilde{\rho}_{j-1;i_{1}i_{2}}-\frac{t_{m_{j}}\tilde{\rho}_{j-1;i_{1}m_{j}}\tilde{\rho}_{j-1;i_{2}m_{j}}}{1+t_{m_{j}}\tilde{\rho}_{j-1;m_{j}m_{j}}},\quad
i_{1},i_{2}\in\mathbb{N}_{0}.$

These recursive definitions match the formulas (56), (57) and (68). Thus, in
effect we are defining the objects associated with an $n$-step confluent
Darboux transformation applied to classical Gegenbauer operators.

######  Proposition 5.8.

Let $i_{1},i_{2},j\in\mathbb{N}_{0}$ and $\tilde{\rho}_{j;i_{1}i_{2}}$,
$\tilde{\pi}_{j,i}$ be as in (95), (94). We have that

(96) $\displaystyle\tilde{\rho}_{j;i_{1}i_{2}}(z)$
$\displaystyle:=\int_{-1}^{z}\tilde{\pi}_{j,i_{1}}(u)\tilde{\pi}_{j,i_{2}}(u)\tilde{W}_{j}(u)du,$
where $\displaystyle\tilde{W}_{j}(z)$
$\displaystyle:=(1-z^{2})^{\alpha-\frac{1}{2}}\tilde{\tau}_{j}(z)^{-2},$

where the integral denotes an anti-derivative that vanishes at $z=-1$

###### Proof.

The proof follows directly from (67) and Proposition 4.11. ∎

######  Proposition 5.9.

Let $\tau^{(\alpha)}_{{\boldsymbol{m}}}$ and
$C^{(\alpha)}_{{\boldsymbol{m}};i}$ be as in (83), (85) and let
$\tilde{\tau}_{n}$, $\tilde{\pi}_{n;i}$ be as in (93), (94) Then,

(97) $\displaystyle\tau^{(\alpha)}_{{\boldsymbol{m}}}$
$\displaystyle=\tilde{\tau}_{n},$ (98) $\displaystyle
C^{(\alpha)}_{{\boldsymbol{m}};i}$ $\displaystyle=\tilde{\pi}_{n;i},\quad
i\in\mathbb{N}_{0}.$

###### Proof.

The proof follows the same argument as the analogous result for exceptional
Legendre polynomials, see [13, Proposition 5]. ∎

As a direct consequence of (97) and (98) we see that the recursively defined
$\tilde{\tau}_{j},\tilde{\pi}_{j;i}$ are polynomials for each $j=1,\ldots,n$.
We have also established that the antiderivative in the RHS of (96) describes
a rational function. This allows us to dispense with the polynomiality and
rationality assumptions used in Section 4.3.

###### Proof of Theorem 5.1.

Starting form the classical Gegenbauer operator, the application of a rational
confluent Darboux transformation indexed by an integer $m_{i}$ introduces an
extra real parameter $t_{m_{i}}$. Proposition 5.9 establishes the equivalence
of the objects defined by the CDT recursion (93), (94) and the matrix-based
definitions (83), (85). This allows us to apply Proposition 4.10. The
eigenvalue relation (86) follows immediately. ∎

###### Proof of Theorem 5.2.

First of all, it is clear by construction that the objects
$C^{(\alpha)}_{{\boldsymbol{m}};i}$ are polynomials. In virtue of Proposition
5.9, the results of Section 4 can be exploited. We recursively define for
$i\in\mathbb{N}_{0}$

$\tilde{\nu}_{0,i}=\nu^{(\alpha)}_{i},\quad\left(\tilde{\nu}_{j,i}\right)^{-1}=\left(\tilde{\nu}_{j-1,i}\right)^{-1}+\delta_{im_{j}}t_{m_{j}},\quad
j=1,\ldots,n.$

By Propositions 4.12 and 4.13, condition (a) is satisfied, i.e.
$\tilde{\tau}_{j}(z)>0$ for $z\in I$, if and only if
$\tilde{\nu}_{j,m_{j}}>0$. By the above definition,

$\tilde{\nu}_{j,m_{j}}=\frac{\nu^{(\alpha)}_{m_{j}}}{1+t_{m_{j}}\nu^{(\alpha)}_{m_{j}}}.$

Therefore, $\tilde{\nu}_{j,m_{j}}>0$ for $j=1,\ldots,n$ if and only if (87)
holds. Relation (88) also follows by Proposition 4.13 and a similar induction
argument. Finally, the completeness condition (c) follows by induction with
Proposition 4.15 serving as the inductive step. ∎

We conclude this section by proving the remaining results on the degrees of
the polynomials and the case of repeated indices.

###### Proof of Proposition 5.3.

By (94), we have

$\tilde{\pi}_{j;m_{j}}=\tilde{\pi}_{j-1,m_{j}},\quad j=1,\ldots,n.$

Identity (92) then follows by (98). Thanks to (92) no generality is lost by
assuming that $i\notin\\{m_{1},\ldots,m_{n}\\}$. We use induction to show that

(99) $\displaystyle\deg\tilde{\tau}_{j}$
$\displaystyle=2(m_{1}+\cdots+m_{j}+\alpha j),\quad j=0,1,\ldots,n.$ (100)
$\displaystyle\deg\tilde{\pi}_{j;i}$ $\displaystyle=\deg\tilde{\tau}_{j}+i.$

The desired relations (90), (91) then follow by (97) and (98).

By inspection, (99) and (100) hold for $j=0$. Assume that these relations hold
for a given $j<n$. By (96),

$\tilde{\rho}_{j,m_{j+1}m_{j+1}}(z)=\int^{z}_{{\color[rgb]{0,0,1}-1}}\left(\frac{\tilde{\pi}_{m_{j+1}}(u)}{\tilde{\tau}_{j}(u)}\right)^{2}\tilde{W}_{0}(u)du.$

Since $m_{j+1}\notin\\{m_{1},\ldots,m_{j}\\}$, by the inductive hypothesis,

$\deg_{z}\tilde{\rho}_{j,m_{j+1}m_{j+1}}=2m_{j+1}+2\alpha,$

where the degree of a rational function is understood as the difference
between the degrees of the numerator and the denominator. Hence,

$\deg_{z}\tilde{\tau}_{j+1}=2m_{j+1}+2\alpha+\deg_{z}\tilde{\tau}_{j},$

which agrees with (99) for the $j+1$ case.

By (94) and (96), we have

$\tilde{\pi}_{j+1;i}=\tilde{\pi}_{j;i}+t_{m_{j+1}}\Pi,$

where

$\Pi(z)=\tilde{\pi}_{j;i}(z)\int^{z}_{-1}\frac{\tilde{\pi}_{j;m_{j+1}}(u)}{\tilde{\tau}_{j}(u)}\frac{\tilde{\pi}_{j;m_{j+1}}(u)}{\tilde{\tau}_{j}(u)}\tilde{W}_{0}(u)du-\tilde{\pi}_{j;m_{j+1}}(z)\int^{z}_{-1}\frac{\tilde{\pi}_{j;i}(u)}{\tilde{\tau}_{j}(u)}\frac{\tilde{\pi}_{j;m_{j+1}}(u)}{\tilde{\tau}_{j}(u)}\tilde{W}_{0}(u)du.$

Since $i\neq m_{j+1}$, the leading degree terms in the above difference do not
cancel and hence

$\deg_{z}\tilde{\pi}_{j+1;i}=\deg_{z}\Pi=2m_{j+1}+2\alpha+\deg\tilde{\pi}_{j;i}.$

Hence (100) also holds for $j+1$. ∎

###### Proof of Proposition 5.7.

We apply Proposition 5.9 and the definitions (93), (94) twice to obtain

$\displaystyle\tau^{(\alpha)}_{({\boldsymbol{m}},j,j)}$
$\displaystyle=\left(1+t^{\prime}_{j}\rho^{(\alpha)}_{({\boldsymbol{m}},j);jj}\right)\tau^{(\alpha)}_{({\boldsymbol{m}},j)}$
$\displaystyle=\left(1+t^{\prime}_{j}\rho^{(\alpha)}_{{\boldsymbol{m}};jj}-\frac{(t^{\prime}_{j})^{2}\left(\rho^{(\alpha)}_{{\boldsymbol{m}};jj}\right)^{2}}{1+t^{\prime}_{j}\rho^{(\alpha)}_{{\boldsymbol{m}};jj}}\right)\left(1+t_{j}\rho^{(\alpha)}_{{\boldsymbol{m}};jj}\right)\tau^{(\alpha)}_{\boldsymbol{m}}$
$\displaystyle=\left(1+(t_{j}+t_{j}^{\prime})\rho^{(\alpha)}_{{\boldsymbol{m}};jj}\right)\tau^{(\alpha)}_{\boldsymbol{m}}$
$\displaystyle=\tau^{(\alpha)}_{({\boldsymbol{m}},j)}(z;{\color[rgb]{0,0,1}(}\boldsymbol{t}_{\boldsymbol{m}},t_{j}+t_{j}^{\prime})),$
$\displaystyle C^{(\alpha)}_{({\boldsymbol{m}},j,j);i}$
$\displaystyle=\left(1+t^{\prime}_{j}\rho^{(\alpha)}_{({\boldsymbol{m}},j);jj}\right)C^{(\alpha)}_{({\boldsymbol{m}},j);i}-t^{\prime}_{j}\rho^{(\alpha)}_{({\boldsymbol{m}},j);ij}C^{(\alpha)}_{({\boldsymbol{m}},j);j}$
$\displaystyle=\left(1+t^{\prime}_{j}\rho^{(\alpha)}_{({\boldsymbol{m}},j);jj}\right)\left(\left(1+t_{j}\rho^{(\alpha)}_{{\boldsymbol{m}};jj}\right)C^{(\alpha)}_{{\boldsymbol{m}};i}-t_{j}\rho^{(\alpha)}_{{\boldsymbol{m}};ij}C^{(\alpha)}_{{\boldsymbol{m}};i}\right)$
$\displaystyle\quad-t^{\prime}_{j}\left(\rho^{(\alpha)}_{{\boldsymbol{m}};ij}-\frac{t_{j}\rho^{(\alpha)}_{{\boldsymbol{m}};jj}\rho^{(\alpha)}_{{\boldsymbol{m}};ij}}{1+t_{j}\rho^{(\alpha)}_{{\boldsymbol{m}};jj}}\right)C^{(\alpha)}_{{\boldsymbol{m}};j}$
$\displaystyle=\left(1+(t_{j}+t_{j}^{\prime})\rho^{(\alpha)}_{{\boldsymbol{m}};jj}\right)C^{(\alpha)}_{{\boldsymbol{m}};i}-(t_{j}+t_{j}^{\prime})\rho^{(\alpha)}_{{\boldsymbol{m}};ij}C^{(\alpha)}_{{\boldsymbol{m}};j}$
$\displaystyle=C^{(\alpha)}_{({\boldsymbol{m}},j);i}(z;(\boldsymbol{t}_{\boldsymbol{m}},t_{j}+t_{j}^{\prime})).$

∎

##  6\. Examples

We conclude by showing some explicit examples of exceptional Gegenbauer
polynomials of the second kind, together with their properties. It can be
readily checked that these families are an isospectral deformation of the
classical Gegenbauer polynomials.

###  6.1. 1-parameter exceptional Gegenbauer polynomials of the second kind

The 1-parameter exceptional Gegenbauer polynomials arise after a single CDT on
the classical operator. For $m\in\mathbb{N}_{0}$, we follow definitions
(56)–(57) to write

$\displaystyle\tau^{(\alpha)}_{m}(z,t_{m})=1+t_{m}\rho^{(\alpha)}_{mm}(z),$
$\displaystyle
C^{(\alpha)}_{m;i}(z,t_{m})=\left(1+t_{m}\rho^{(\alpha)}_{mm}(z)\right)C^{(\alpha)}_{i}(z)-t_{m}\rho^{(\alpha)}_{im}(z)C^{(\alpha)}_{m}(z),$

with $\rho^{(\alpha)}_{ij}$ as per (81) and $C^{(\alpha)}_{i}$ the classical
Gegenbauer polynomials. Notice that by construction, we have
$C^{(\alpha)}_{m;m}=C^{(\alpha)}_{m}$. The
$\\{C^{(\alpha)}_{m;i}\\}_{i\in\mathbb{N}_{0}}$ is a family of exceptional
Gegenbauer polynomials with weight

$W^{(\alpha)}_{m}(z)=\frac{(1-z^{2})^{\alpha-\frac{1}{2}}}{\big{(}\tau^{(\alpha)}_{m}(z)\big{)}^{2}},$

as long as $t_{m}$ satisfies the inequality

$\displaystyle
t_{m}>-\left(\nu^{(\alpha)}_{m}\right)^{-1}=-\frac{2^{2\alpha-1}m!(m+\alpha)\Gamma(\alpha)^{2}}{\pi\Gamma(m+2\alpha)}.$

The orthogonality relations are

$\displaystyle\int_{-1}^{1}C^{(\alpha)}_{m;i}(z,t_{m})C^{(\alpha)}_{m;j}(z,t_{m})W^{(\alpha)}_{m}(z)dz=\delta_{ij}\nu^{(\alpha)}_{i}=\delta_{ij}\frac{\pi
2^{1-2\alpha}\Gamma(i+2\alpha)}{i!(i+\alpha)\Gamma(\alpha)^{2}},\quad
i,j\in\mathbb{N}_{0}\backslash\\{m\\}$
$\displaystyle\int_{-1}^{1}C^{(\alpha)}_{m;m}(z,t_{m})^{2}W^{(\alpha)}_{m}(z,t_{m})dz=\frac{\nu^{(\alpha)}_{m}}{1+t\nu^{(\alpha)}_{m}}=\frac{\pi
2^{1-2\alpha}\Gamma(m+2\alpha)}{m!(m+\alpha)\Gamma(\alpha)^{2}+t_{m}\pi
2^{1-2\alpha}\Gamma(m+2\alpha)}.$

The function $\tau^{(\alpha)}_{m}$ is shown below in Figure 1, for a
particular choice of the parameters $\alpha,m,t_{m}$.

Figure 1. The function $\tau^{(\alpha)}_{m}(z,t_{m})$ for $m=4$ and
$t_{m}=0.5$.

Clearly, there are no zeroes. Figure 2 shows the polynomial families for
$\alpha=3/2$, $m=2$, and different values of $t_{m}$.

|
---|---
Figure 2. First few exceptional Gegenbauer polynomials
$C^{(\alpha)}_{m;i}(z;t_{m})$ for $m=4$, with $t_{m}=0$ (left) and $t_{m}=0.5$
(right).

The first few polynomials for $m=4$ and $\alpha=3/2$ are explicitly given by

$\displaystyle
C^{(3/2)}_{4;0}(z,t_{4})=C^{(3/2)}_{0}(z)+\frac{15}{176}t_{4}\left(945z^{11}-3080z^{9}+3630z^{7}-1848z^{5}+385z^{3}+32\right),$
$\displaystyle
C^{(3/2)}_{4;1}(z,t_{4})=C^{(3/2)}_{1}(z)+\frac{45}{5632}t_{4}\big{(}19845z^{12}-59290z^{10}+59455z^{8}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-20636z^{6}+275z^{4}+1430z^{2}+1024z-55\big{)},$
$\displaystyle
C^{(3/2)}_{4;2}(z,t_{4})=C^{(3/2)}_{2}(z)+\frac{45}{704}t_{4}\big{(}3675z^{13}-11515z^{11}+13310z^{9}-7590z^{7}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+2871z^{5}-495z^{3}+320z^{2}-64\big{)},$
$\displaystyle
C^{(3/2)}_{4;3}(z,t_{4})=C^{(3/2)}_{3}(z)+\frac{75}{2816}t_{4}\big{(}9261z^{14}-30919z^{12}+39501z^{10}-24783z^{8}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+7007z^{6}+2739z^{4}+1792z^{3}-1881z^{2}-768z+99\big{)},$
$\displaystyle C^{(3/2)}_{4;4}(z,t_{4})=C^{(3/2)}_{4}(z).$

Notice that, as illustrated by Figure 2, the exceptional polynomials are
continuous deformations of the corresponding classical polynomials, aside from
$C^{(3/2)}_{4;4}$.

###  6.2. 2-parameter exceptional Gegenbauer polynomials of the second kind

We construct the 2-parameter exceptional polynomials by applying the recursive
construction to the 1-parameter formulas. We start with
${\boldsymbol{m}}=(m_{1},m_{2})\in\mathbb{N}_{0}^{2}$ and the associated tuple
$\boldsymbol{t}_{{\boldsymbol{m}}}=(t_{m_{1}},t_{m_{2}})$. Following equations
(55)–(57), and using (68), we find that

$\displaystyle\tau^{(\alpha)}_{{\boldsymbol{m}}}(z;\boldsymbol{t}_{{\boldsymbol{m}}})=\tau^{(\alpha)}_{m_{1}}(z,t_{m_{1}})\tau^{(\alpha)}_{m_{2}}(z,t_{m_{2}})-t_{m_{1}}t_{m_{2}}\rho^{(\alpha)}_{m_{1}m_{2}}(z)^{2},$
$\displaystyle
C^{(\alpha)}_{{\boldsymbol{m}};i}(z;\boldsymbol{t}_{{\boldsymbol{m}}})=C^{(\alpha)}_{i}(z)\tau^{(\alpha)}_{{\boldsymbol{m}}}(z;\boldsymbol{t}_{{\boldsymbol{m}}})-t_{m_{1}}C^{(\alpha)}_{m_{1}}(z)\tau^{(\alpha)}_{m_{2}}(z;t_{m_{2}})\rho^{(\alpha)}_{m_{2};m_{1}i}(z,t_{m_{2}})$
$\displaystyle\qquad\qquad\qquad-
t_{m_{2}}C^{(\alpha)}_{m_{2}}(z)\tau^{(\alpha)}_{m_{1}}(z;t_{m_{2}})\rho^{(\alpha)}_{m_{1};m_{2}i}(z,t_{m_{1}}),$

where

$\displaystyle\rho^{(\alpha)}_{m;ij}(z,t_{m})$
$\displaystyle=\int_{-1}^{z}C^{(\alpha)}_{m;i}(u,t_{m})C^{(\alpha)}_{m;j}(u,t_{m})W^{(\alpha)}_{m}(u)du$
$\displaystyle=\rho^{(\alpha)}_{ij}(z)-\frac{t_{m}\rho^{(\alpha)}_{mi}(z)\rho^{(\alpha)}_{mj}(z)}{1+t_{m}\rho^{(\alpha)}_{mm}(z)},\qquad
m,i,j\in\mathbb{N}_{0}.$

Once again, the polynomials form a complete orthogonal basis relative to the
weight

$W^{(\alpha)}_{\boldsymbol{m}}(z)=(1-z^{2})^{\alpha-\frac{1}{2}}\tau^{(\alpha)}_{m_{1}m_{2}}(z;t_{m_{1}},t_{m_{2}})^{-2},$

provided

$t_{m_{i}}>-\frac{2^{2\alpha-1}m_{i}!(m_{i}+\alpha)\Gamma(\alpha)^{2}}{\pi\Gamma(m_{i}+2\alpha)},\quad
i=1,2.$

The orthogonality relations are

$\displaystyle\int_{-1}^{1}C^{(\alpha)}_{{\boldsymbol{m}};i}(z,\boldsymbol{t}_{{\boldsymbol{m}}})C^{(\alpha)}_{{\boldsymbol{m}};j}(z,\boldsymbol{t}_{{\boldsymbol{m}}})W^{(\alpha)}_{\boldsymbol{m}}(z,\boldsymbol{t}_{\boldsymbol{m}})dz=\delta_{ij}\frac{\pi
2^{1-2\alpha}\Gamma(i+2\alpha)}{i!(i+\alpha)\Gamma(\alpha)^{2}},\quad
i,j\in\mathbb{N}_{0}\backslash\\{m_{1},m_{2}\\},$
$\displaystyle\int_{-1}^{1}C^{(\alpha)}_{{\boldsymbol{m}};m_{i}}(z,\boldsymbol{t}_{{\boldsymbol{m}}})^{2}W^{(\alpha)}_{\boldsymbol{m}}(z,\boldsymbol{t}_{\boldsymbol{m}})dz=\frac{\pi
2^{1-2\alpha}\Gamma(m_{i}+2\alpha)}{m_{i}!(m_{i}+\alpha)\Gamma(\alpha)^{2}+t_{m_{i}}\pi
2^{1-2\alpha}\Gamma(m_{i}+2\alpha)},\quad i=1,2.$

### Acknowledgements

María Ángeles García-Ferrero was supported by the Spanish MINECO through Juan
de la Cierva fellowship FJC2019-039681-I, by the Spanish State Research Agency
through BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and by the
Basque Government through the BERC Programme 2018-2021. The research of David
Gómez-Ullate has been financed by projects PGC2018-096504-B-C33 and
RTI2018-100754-B-I00 from FEDER/Ministerio de Ciencia e Innovación– Agencia
Estatal de Investigación, the European Union under the 2014-2020 ERDF
Operational Programme and the Department of Economy, Knowledge, Business and
University of the Regional Government of Andalusia (project FEDER-
UCA18-108393).

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