# Epilegomena to the study of semiclassical orthogonal polynomials

K. Castillo  CMUC, Department of Mathematics, University of Coimbra, 3001-501
Coimbra, Portugal<EMAIL_ADDRESS>and  D. Mbouna  Department of Mathematics,
Faculty of Sciences, University of Porto, Campo Alegre st., 687, 4169-007
Porto, Portugal<EMAIL_ADDRESS>
###### Abstract.

In his monograph [Classical and quantum orthogonal polynomials in one
variable, Cambridge University Press, 2005 (paperback edition 2009)], Ismail
conjectured that certain structure relations involving the Askey-Wilson
operator characterize proper subsets of the set of all
$\mathcal{D}_{q}$-classical orthogonal polynomials, here to be understood as
the Askey-Wilson polynomials and their limit cases. In this paper we give two
characterization theorems for $\mathcal{D}_{q}$-semiclassical (and classical)
orthogonal polynomials in consonance with the pioneering works by Maroni [Ann.
Mat. Pura. Appl. (1987)] and Bonan, Lubinsky, and Nevai [SIAM J. Math. Anal.
18 (1987)] for the standard derivative, re-establishing in this context the
perfect “symmetry” between the standard derivative and the Askey-Wilson
operator. As an application, we present a sequence of
$\mathcal{D}_{q}$-semiclassical orthogonal polynomials of class two that
disproves Ismail’s conjectures. Further results are presented for Hahn’s
operator.

###### 2010 Mathematics Subject Classification:

33D45

##  1\. Introduction

The term semiclassical for a special class of orthogonal polynomials was
coined in 1984 by Hendriksen and van Rossum (see [13]) during the Laguerre
Symposium held at Bar-le-Duc, whose main speaker was Dieudonné. However, in
the first line of his monumental work entitled “Une théorie algébrique des
polynômes orthogonaux. Applications aux polynômes orthogonaux semi-classiques”
(see [25]), Maroni referred to Shohat111On his recent visit to Portugal, C.
Brezinski showed us the only known photo of J. Shohat, which will appear in
his book on the history of orthogonal polynomials written in collaboration
with M. Redivo-Zaglia. (see [28]) as “l’inventeur des polynômes semi-
classiques”. It was at the hands of Maroni that semiclassical orthogonal
polynomials have become such a highly developed topic, although, as he himself
points out, these sequences of orthogonal polynomials (OP) have always been
present in certain structure relations which are as old as orthogonal
polynomials themselves. Let us recall one of the best known problems in this
regard. According to Al-Salam and Chihara (see [2, p. 69]), Askey raised the
question of characterizing OP, $(P_{n})_{n\geq 0}$, satisfying

(1)
$\displaystyle\phi\,P^{\prime}_{n}=\sum_{j=-M}^{N}a_{n,j}P_{n+j}\qquad(c_{n,j}\in\mathbb{C};\,M,N\in\mathbb{N}),$

$\phi$ being a polynomial which does not depend on $n$. In [2] it was proved
that the only OP that satisfy (1) for $M=N=1$ are the old classical
polynomials, i.e., Jacobi, Bessel, Hermite, and Laguerre polynomials. However,
for arbitrary values of $M$ and $N$, there are nonclassical OP for which (1)
holds. For instance, in 1984 Koornwinder considered (see [18]), without
explicit mention of this, OP, $(P_{n}^{\alpha,\beta,M_{0},M_{1}})_{n\geq 0}$,
satisfying (1), whose weight function is a linear combination of the Jacobi
measure and two mass points at $-1$ and $1$:
$(1-x)^{\alpha}(1+x)^{\beta}+M_{0}\delta(x+1)+M_{1}\delta(x-1)$
($\alpha,\beta>-1;M_{0},M_{1}\in\mathbb{N}$). Indeed, as Koornwinder recently
proved in a private communication (see also [19]),

$\displaystyle\int_{-1}^{1}(1-x^{2})^{2}\,\frac{\mathrm{d}P_{n}^{\alpha,\beta,M_{0},M_{1}}}{\mathrm{d}x}(x)Q(x)(1-x)^{\alpha}(1+x)^{\beta}\mathrm{d}x$
$\displaystyle\quad+M_{0}(1-x^{2})^{2}\,\left.\frac{\mathrm{d}P_{n}^{\alpha,\beta,M_{0},M_{1}}}{\mathrm{d}x}(x)Q(x)\right|_{x=1}$
$\displaystyle\quad+M_{1}(1-x^{2})^{2}\,\left.\frac{\mathrm{d}P_{n}^{\alpha,\beta,M_{0},M_{1}}}{\mathrm{d}x}(x)Q(x)\right|_{x=-1}=0,$

$Q$ being a polynomial of degree at most $n-4$. Hence

$(1-x^{2})^{2}\,\frac{\mathrm{d}P_{n}^{\alpha,\beta,M_{0},M_{1}}}{\mathrm{d}x}(x)=\sum_{j=-3}^{3}c_{n,j}P_{n+j}^{\alpha,\beta,M_{0},M_{1}}(x)\qquad(c_{n,j}\in\mathbb{C}).$

In 1985 (see [24, Theorem 3.1] and [21, Theorem 3.1]), Maroni, in his
prolegomena to the study of semiclassical orthogonal polynomials, answers
Askey’s question by characterizing semiclassial OP222According to Maroni (see
[25, Definition 7.1]), the classical orthogonal polynomials are semiclassical
of class $0$. However, subsequently, when we refer to semiclassical orthogonal
polynomials, we exclude the classical ones. (see also [22]). (Note that from
[24, Theorem 3.1] and [23, Theorem 3.1] (see also [15]), it follows
immediately that Koornwinder’s polynomials are semiclassical OP satisfying
(1).) In the same year, Bonan, Lubinsky, and Nevai solved a more general
problem using analytic methods in the framework of orthogonality in the
positive definite sense (see [5, Theorem 1.1]). Although much has been written
about this, there are still outstanding problems that remain open. For
instance, we may rewrite Askey’s question by changing the standard derivative
by the Askey-Wilson operator. Recall that usually the Askey-Wilson divided
difference operator $\mathcal{D}_{q}:\mathcal{P}\to\mathcal{P}$ is defined by

(2)
$\displaystyle\mathcal{D}_{q}\,f(x)=\frac{\breve{f}\left(q^{1/2}e^{i\theta}\right)-\breve{f}\big{(}q^{-1/2}e^{i\theta}\big{)}}{\breve{e}\big{(}q^{1/2}e^{i\theta}\big{)}-\breve{e}\big{(}q^{-1/2}e^{i\theta}\big{)}},$

where $\breve{f}(e^{i\theta})=f(\cos\theta)$ for each polynomial $f$,
$e(x)=x$, and $\theta$ is not necessarily a real number (see [14, Section
12.1]). Here and subsequently, $0<q<1$ is assumed fixed. ($\mathcal{P}$
denotes the vector space of all polynomials with complex coefficients.) Taking
$e^{i\theta}=q^{s}$ in (2), $\mathcal{D}_{q}$ reads

$\mathcal{D}_{q}\,f(x(s))=\frac{f(x(s+1/2))-f(x(s-1/2))}{x(s+1/2)-x(s-1/2)},$

with $x(s)=(q^{s}+q^{-s})/2$. What is expected by most of the people working
on the subject, in contradiction with the case of the standard derivative
operator, is that the answer to this and other related questions be proper
subsets of the set of all $\mathcal{D}_{q}$-classical orthogonal polynomials,
here to be understood as the Askey-Wilson polynomials and their limit cases.
Recall that the Askey-Wilson polynomials (see [17, Section 14.1])

$p_{n}(x;a,b,c,d\,|\,q)=a^{-n}\,(ab,ac,ad;q)_{n}\,{}_{4}\phi_{3}\left(\left.\genfrac{.}{.}{0.0pt}{}{q^{-n}\mathchar
44\relax\mkern 6.0muabcbdq^{n-1}\mathchar 44\relax\mkern
6.0muae^{i\theta}\mathchar 44\relax\mkern 6.0muae^{-i\theta}}{ab\mathchar
44\relax\mkern 6.0muac\mathchar 44\relax\mkern 6.0muad}\right|\,q\mathchar
44\relax\mkern 6.0muq\right),$

where $x=\cos\theta$, are the $q$-analogues of the Wilson polynomials. (If we
take $a=q^{1/2\alpha+1/4}$, $b=q^{1/2\alpha+3/4}$, $c=-a$, and $d=-b$, we get
the continuous $q$-Jacobi polynomials. If we take $c=d=0$, we get the Al-
Salam-Chihara polynomials.) In this sense, there are in the literature two
well-known conjectures posed by Ismail (see [14, Conjecture 24.7.8] and [14,
Conjecture 24.7.9]).

######  Conjecture 1.1.

Let $(P_{n})_{n\geq 0}$ be a sequence of orthogonal polynomials and let $\phi$
be a polynomial which does not depend on $n$. If

(3)
$\displaystyle\phi\,\mathcal{D}_{q}\,P_{n}=\sum_{j=-1}^{1}a_{n,j}P_{n+j}\qquad(a_{n,j}\in\mathbb{C}),$

then $P_{n}$ is a multiple of the continuous $q$-Jacobi polynomials or Al-
Salam-Chihara polynomials, or special or limiting cases of them. The same
conclusion holds if

(4)
$\displaystyle\phi\,\mathcal{D}_{q}\,P_{n}=\sum_{j=-M}^{N}a_{n,j}P_{n+j}\qquad(a_{n,j}\in\mathbb{C};\,M,N\in\mathbb{N}).$

######  Conjecture 1.2.

Let $(P_{n})_{n\geq 0}$ be a sequence of orthogonal polynomials and $\phi$ be
a polynomial of degree at most $4$. Then $(P_{n})_{n\geq 0}$ satisfies

(5)
$\displaystyle\phi\,\mathcal{D}_{q}^{2}P_{n}=\sum_{j=-M}^{N}a_{n,j}P_{n+j}\qquad(a_{n,j}\in\mathbb{C};\,M,N\in\mathbb{N}),$

if and only if $P_{n}$ is a multiple of $p_{n}(x;a,b,c,d\,|\,q)$ for some for
some parameters $a,b,c,d$.

In [1], Al-Salam proved Conjecture 1.1 for $\phi=1$ by characterizing the
continuous $q$-Hermite polynomials (see [17, Section 14.26]). In [8], we prove
that the Al-Salam Chihara polynomials (see [17, Section 14.26]), with nonzero
parameters $a$ and $b$ such that $a/b=q^{\pm 1/2}$, are the only OP satisfying
(3) for $\deg\phi=1$. We also prove that the Chebyschev polynomials of the
first kind and the continuous $q$-Jacobi polynomials (see [17, Section 14.10])
are the only ones satisfying (3) for $\deg\phi=2$. Moreover, in [6,
Proposition 2.1] we prove that the continuous dual $q$-Hahn polynomials (see
[17, Section 14.3]), with parameters $a=1$, $b=-1$, $c=q^{1/4}$, and $q$
replaced by $q^{1/2}$, satisfy (4) with $M=2$ and $N=1$, which disproves the
second part of Conjecture 1.1. On the other hand, Conjecture 1.2 is claimed to
be positively solved in [16], but the authors only proved, partially, the case
$M=2$ and $N=2$ 333It is worth pointing out that the main result of [16] is a
particular case of a direct consequence of Sonine-Hahn’s theorem for the
Askey-Wilson operator (see remark after [7, Theorem 1.2]).. Recently we noted
that above conjectures are related with the theory of
$\mathcal{D}_{q}$-semiclassical orthogonal polynomials. Indeed, in Section 3,
we characterize $\mathcal{D}_{q}$-semiclassical (and classical) orthogonal
polynomials from the structure relation (4), in consonance with the works by
Maroni [21] and Bonan, Lubinsky, and Nevai [5] for the standard derivative,
re-establishing in this context the perfect “symmetry” between the standard
derivative and the Askey-Wilson operator. As an application of these results,
in Section 4, we present an example of $\mathcal{D}_{q}$-semiclassical
orthogonal polynomials that disproves Conjecture 1.1. In Section 4 we also
show that the OP that disproves Conjecture 1.1 also disproves Conjecture 1.2.
Finally, in Section 5, we explore our ideas when instead of the Askey-Wilson
operator we consider the Hahn operator, which is related with the structure
relation of another conjecture posed by Ismail (see [14, Conjecture 24.7.7]),
but first some preliminary definitions and basic results are needed.

##  2\. Preliminary results

Let $\mathcal{P}^{*}$ be the set of all linear forms on $\mathcal{P}$ and let
$\mathcal{P}_{n}$ be the subspace of $\mathcal{P}$ of all polynomials with
degree less than or equal to $n$. Set $\mathcal{P}_{-1}=\\{0\\}$. A free
system in $\mathcal{P}$ is a sequence $(Q_{n})_{n\geq 0}$ such that
$Q_{n}\in\mathcal{P}_{n}\setminus\mathcal{P}_{n-1}$ for each $n$. A free
system $(P_{n})_{n\geq 0}$ is called OP with respect to ${\bf
u}\in\mathcal{P}^{*}$ if

$\langle{\bf
u},P_{n}P_{m}\rangle=h_{n}\delta_{n,m}\quad(m=0,1,\ldots;\,h_{n}\in\mathbb{C}\setminus\\{0\\}),$

$\langle{\bf u},f\rangle$ being the action of ${\bf u}$ on $f$. ${\bf u}$ is
called regular if there exists an OP with respect to it. Recall that a (monic)
OP, $(P_{n})_{n\geq 0}$, satisfies the following recurrence relation:

(6) $\displaystyle
xP_{n}(x)=P_{n+1}(x)+B_{n}P_{n}(x)+C_{n}P_{n-1}(x)\qquad(B_{n}\in\mathbb{C},\,C_{n+1}\in\mathbb{C}\setminus\\{0\\}),$

with initial conditions $P_{-1}=0$ and $P_{0}=1$. Hence it follows that

$B_{n}=\frac{\langle\mathbf{u},xP^{2}_{n}(x)\rangle}{\langle\mathbf{u},P^{2}_{n}(x)\rangle},\qquad
C_{n}=\frac{\langle\mathbf{u},P^{2}_{n}\rangle}{\langle\mathbf{u},P^{2}_{n-1}\rangle}.$

(Of course, there is no loss of generality in assuming $C_{0}=0$.) Since the
elements of $\mathcal{P}^{*}$ are completely determined by its action on a
system of generators of $\mathcal{P}$, we say that $\mathbf{u}=\mathbf{v}$ if
and only if

$\mathbf{u}_{n}=\left\langle{\mathbf{u},x^{n}}\right\rangle=\left\langle{\mathbf{v},x^{n}}\right\rangle$

for all $n\in\mathbb{N}$. In the set $\mathcal{P}^{*}$, addition and
multiplications by scalars can be defined by

$\displaystyle\left\langle{\mathbf{u}+\mathbf{v},x^{n}}\right\rangle$
$\displaystyle=\left\langle{\mathbf{u},x^{n}}\right\rangle+\left\langle{\mathbf{v},x^{n}}\right\rangle,$
$\displaystyle\left\langle{c\mathbf{u},x^{n}}\right\rangle$
$\displaystyle=c\left\langle{\mathbf{u},x^{n}}\right\rangle\qquad(c\in\mathbb{C}),$

for all $n\in\mathbb{N}$. $\mathcal{P}^{*}$, endowed with these operations, is
a vector space over $\mathcal{P}$. In $\mathcal{P}^{*}$, the identity for the
additivity is denoted by $\mathbf{0}$ and called the zero. The zero is
therefore defined by the relation
$\left\langle{\mathbf{0},x^{n}}\right\rangle=0$ for all $n\in\mathbb{N}$. Note
that $f\,\mathbf{u}=g\,\mathbf{u}=\mathbf{0}$ $(f,g\in\mathcal{P})$ if and
only if $\mathbf{u}=\mathbf{0}$. The left multiplication of ${\bf u}$ by
$f\in\mathcal{P}$, denoted by $f{\bf u}:\mathcal{P}\to\mathcal{P}$, is the
form defined by

$\langle f{\bf u},x^{n}\rangle=\langle{\bf u},fx^{n}\rangle,$

for all $n\in\mathbb{N}$. The division of $\mathbf{u}$ by a polynomial,
denoted by $(x-c)^{-1}\mathbf{u}:\mathcal{P}\to\mathcal{P}$, is the form
defined by

$\left\langle{(x-c)^{-1}\mathbf{u},f}\right\rangle=\left\langle{\mathbf{u},\frac{f(x)-f(c)}{x-c}}\right\rangle\qquad(c\in\mathbb{C};\,f\in\mathcal{P}).$

Define also $\mathbf{\delta}_{c}:\mathcal{P}\to\mathbb{C}$ by
$\mathbf{\delta}_{c}f(x)=f(c)$. We check at once that

$(x-c)((x-c)^{-1}\mathbf{u})=\mathbf{u},\qquad(x-c)^{-1}((x-c)\mathbf{u})=\mathbf{u}-\mathbf{u}_{0}\,\delta_{c}.$

$\mathcal{P}$ may be endowed with an appropriate strict inductive limit
topology such that the algebraic and the topological dual spaces of
$\mathcal{P}$ coincide (see [29, Chapter 13]), that is,

(7) $\displaystyle\mathcal{P}^{*}=\mathcal{P}^{\prime}.$

Given a free system $(Q_{n})_{n\geq 0}$, the corresponding dual basis is a
sequence of linear forms ${\bf a}_{n}:\mathcal{P}\to\mathbb{C}$ such that

$\langle{\bf a}_{n},Q_{m}\rangle=\delta_{n,m},$

and so, for an OP $(P_{n})_{n\geq 0}$, ${\bf a}_{n}$ is explicitly given by

${\bf a}_{n}=\frac{P_{n}}{\langle{\bf a}_{n},P^{2}_{n}\rangle}\mathbf{u}.$

In the sense of the weak dual topology, it would be easy to explicitly build
bases in dual space. Indeed,

$\mathbf{u}=\sum_{j=0}^{\infty}\left\langle{\mathbf{u},Q_{j}}\right\rangle\mathbf{a}_{j}\qquad(\mathbf{u}\in\mathcal{P}^{\prime});$

this will be essential in the sequel. For more details we refer the reader to
[25] (see also [10]).

The Askey-Wilson average operator $\mathcal{S}_{q}:\mathcal{P}\to\mathcal{P}$
is defined by (see [14, p. 301])

$\displaystyle\mathcal{S}_{q}f(x(s))=\frac{f\big{(}x(s+1/2)\big{)}+f\big{(}x(s-1/2)\big{)}}{2}.$

for every polynomial $f$. It is easy to see that
$\mathcal{D}_{q}\,x^{n}=\gamma_{n}x^{n-1}+(\text{lower degree terms})$ and
$\mathcal{S}_{q}\,x^{n}=\alpha_{n}x^{n}+(\text{lower degree terms})$ for all
$n\in\mathbb{N}$, where we have set

(8) $\displaystyle\alpha_{n}$
$\displaystyle=\frac{q^{n/2}+q^{-n/2}}{2},\qquad\gamma_{n}=\frac{q^{n/2}-q^{-n/2}}{q^{1/2}-q^{-1/2}}.$

Set $\gamma_{-1}=-1$ and $\alpha_{-1}=\alpha$. For every ${\bf
u}\in\mathcal{P^{*}}$ and $f\in\mathcal{P}$,
$\mathbf{D}_{q}:\mathcal{P}^{*}\to\mathcal{P}^{*}$ and
$\mathbf{S}_{q}:\mathcal{P}^{*}\to\mathcal{P}^{*}$ are defined by
transposition:

$\displaystyle\langle\mathbf{D}_{q}{\bf u},f\rangle=-\langle{\bf
u},\mathcal{D}_{q}f\rangle,\qquad\langle\mathbf{S}_{q}{\bf
u},f\rangle=\langle{\bf u},\mathcal{S}_{q}f\rangle.$

The next definition extends the definition of classical linear forms given by
Geronimus [12] and Maroni (see [26, Proposition 2.1]).

######  Definition 2.1.

[9, Definition 3.1] ${\bf u}\in\mathcal{P}^{*}$ is called
$\mathbf{D}_{q}$-classical if it is regular and there exist
$\phi\in\mathcal{P}_{2}\setminus\mathcal{P}_{-1}$ and
$\psi\in\mathcal{P}_{1}\setminus\mathcal{P}_{-1}$ such that

(9) $\displaystyle\mathbf{D}_{q}(\phi{\bf u})=\mathbf{S}_{q}(\psi{\bf u}).$

$($We will call it simply classical when no confusion can arise.$)$

Observe that (9) condenses all the information of a sequence of classical
orthogonal polynomials in the first three non-constant polynomials of said
sequence. The next theorem gives tractable necessary and sufficient conditions
for the existence of solutions of (9), characterizing the linear form
$\mathbf{u}$ and, in particular, solving the question of the existence of
classical OP.

######  Theorem 2.1.

[9, Theorem 4.1] Suppose that ${\bf u}\in\mathcal{P}^{*}$ satisfies (9) with
$\phi(x)=ax^{2}+bx+c$ and $\psi(x)=dx+e$. Then ${\bf u}$ is regular if and
only if

$\displaystyle d_{n}\neq
0,\qquad\phi^{[n]}\left(-\frac{e_{n}}{d_{2n}}\right)\neq 0,$

for all $n\in\mathbb{N}$, where $d_{n}=a\gamma_{n}+d\alpha_{n}$,
$e_{n}=b\gamma_{n}+e\alpha_{n}$, $\alpha_{n}$ and $\gamma_{n}$ being defined
by (8), and

$\displaystyle\phi^{[n]}(x)=\big{(}d(\alpha^{2}-1)\gamma_{2n}+a\alpha_{2n}\big{)}\big{(}x^{2}-1/2\big{)}+\big{(}b\alpha_{n}+e(\alpha^{2}-1)\gamma_{n}\big{)}x+c+a/2.$

OP with respect to ($\mathbf{D}_{q}$-)classical linear forms are called
($\mathcal{D}_{q}$-)classical polynomials. Unlike when dealing with the
standard derivative, in the case of Definition 2.1 it is still an open problem
to describe the solutions of (9) (see [10, Theorem 3.2]).

######  Theorem 2.2.

[14, Theorem 20.1.3] The equation

(10) $\displaystyle
f(x)\mathcal{D}^{2}_{q}\,y+g(x)\mathcal{S}_{q}\mathcal{D}_{q}\,y+h(x)\,y=\lambda_{n}\,y$

has a polynomial solution $P_{n}\in\mathcal{P}_{n}\setminus\mathcal{P}_{n-1}$
if and only if $P_{n}$ is a multiple of $p_{n}(x;a,b,c,d\,|\,q)$ for some
parameters $a,b,c,d$ including limiting cases as one or more of the parameters
tends to $\infty$. In all these cases $f$, $g$, $h$, and $\lambda_{n}$ reduce
to

$\displaystyle f(x)$
$\displaystyle=-q^{-1/2}(2(1+\sigma_{4})x^{2}-(\sigma_{1}+\sigma_{3})x-1+\sigma_{2}-\sigma_{4}),$
$\displaystyle g(x)$
$\displaystyle=\frac{2}{1-q}(2(\sigma_{4}-1)x+\sigma_{1}-\sigma_{3}),\qquad
h(x)=0,$ $\displaystyle\lambda_{n}$
$\displaystyle=\frac{4q(1-q^{-n})(1-\sigma_{4}q^{n-1})}{(1-q)^{2}},$

or a special or limiting case of it, $\sigma_{j}$ being the jth elementary
symmetric function of the Askey-Wilson parameters.

Let us recall some useful operations.

######  Lemma 2.1.

[9, Lemma 2.1] Let $f,g\in\mathcal{P}$ and ${\bf u}\in\mathcal{P}^{*}$. Then
the following hold:

(11) $\displaystyle\mathcal{D}_{q}\big{(}fg\big{)}$
$\displaystyle=\big{(}\mathcal{D}_{q}f\big{)}\big{(}\mathcal{S}_{q}g\big{)}+\big{(}\mathcal{S}_{q}f\big{)}\big{(}\mathcal{D}_{q}g\big{)},$
(12) $\displaystyle\mathcal{S}_{q}(fg)$
$\displaystyle=(\mathcal{D}_{q}f)(\mathcal{D}_{q}g)\texttt{U}_{2}+(\mathcal{S}_{q}f\big{)}\big{(}\mathcal{S}_{q}g),$
(13) $\displaystyle f\mathcal{D}_{q}g$
$\displaystyle=\mathcal{D}_{q}\left((\mathcal{S}_{q}f-\alpha^{-1}\texttt{U}_{1}\mathcal{D}_{q}f)g\right)-\alpha^{-1}\mathcal{S}_{q}(g\mathcal{D}_{q}f),$
(14) $\displaystyle\alpha{\bf D}_{q}(f{\bf u})$
$\displaystyle=\left(\alpha\mathcal{S}_{q}f-\texttt{U}_{1}\mathcal{D}_{q}f\right){\bf
D}_{q}{\bf u}+\mathcal{D}_{q}f{\bf S}_{q}{\bf u},$ (15)
$\displaystyle\alpha{\bf S}_{q}(f{\bf u})$
$\displaystyle=(\alpha^{2}\texttt{U}_{2}-\texttt{U}_{1}^{2})\mathcal{D}_{q}f~{}{\bf
D}_{q}{\bf u}+(\alpha\mathcal{S}_{q}f+\texttt{U}_{1}\mathcal{D}_{q}f){\bf
S}_{q}{\bf u},$ (16) $\displaystyle f{\bf D}_{q}{\bf u}$ $\displaystyle={\bf
D}_{q}\left(\mathcal{S}_{q}f~{}{\bf u}\right)-{\bf
S}_{q}\left(\mathcal{D}_{q}f~{}{\bf u}\right),$ (17) $\displaystyle f{\bf
S}_{q}{\bf u}$ $\displaystyle={\bf S}_{q}\left(\mathcal{S}_{q}f~{}{\bf
u}\right)-{\bf D}_{q}\left(\texttt{U}_{2}\mathcal{D}_{q}f~{}{\bf u}\right),$
(18) $\displaystyle\alpha\mathbf{D}_{q}^{n}\mathbf{S}_{q}{\bf u}$
$\displaystyle=\alpha_{n+1}\mathbf{S}_{q}\mathbf{D}_{q}^{n}{\bf
u}+\gamma_{n}\texttt{U}_{1}\mathbf{D}_{q}^{n+1}{\bf u}\qquad(n\in\mathbb{N}),$

where $\texttt{U}_{1}(x)=(\alpha^{2}-1)x$ and
$\texttt{U}_{2}(x)=(\alpha^{2}-1)(x^{2}-1).$

The following result clarifies which are the $\mathcal{D}_{q}$-classical
orthogonal polynomials.

######  Proposition 2.1.

The $\mathcal{D}_{q}$-classical sequences of orthogonal polynomials are the
sequences of Askey-Wilson polynomials
$(p_{n}(x;a,b,c,d\,|\,q))_{n=0}^{\infty}$ for some parameters $a,b,c,d$
including limiting cases as one or more of the parameters tends to $\infty$.

###### Proof.

This follows from Theorem 2.2, after showing the equivalence between (9) and
(10) with $h=0$ (see [11, Theorem 5]). (The interested reader can also prove
this easily following the proof of [26, Proposition 2.8].) ∎

From Definition 2.1 we introduce $\mathbf{D}_{q}$-semiclassical linear forms
in a natural way (see, for instance, [26, Section 3]).

######  Definition 2.2.

We call a linear form, in $\mathcal{P}^{*}$, $\mathbf{D}_{q}$-semiclassical if
it is regular, not $\mathbf{D}_{q}$-classical, and there exist two polynomials
$\phi$ and $\psi$ with at least one of them nonzero, such that (9) holds.
$($We will call it simply semiclassical when no confusion can arise.$)$

Under the conditions of Definition 2.2, necessarily both $\phi$ and $\psi$ are
not zero and $\deg\psi\geq 1$. The class of ${\bf u}$ is the positive integer

$\displaystyle s=\min_{(\phi,\psi)\in\mathcal{P}_{\bf u}}\max\\{\mbox{\rm
deg}\,\phi-2,\mbox{\rm deg}\,\psi-1\\},$

where $\mathcal{P}_{\bf u}$ is the set of all pairs $(\phi,\psi)$ of nonzero
polynomials such that (9) holds. (Note that when $s=0$ we have the classical
linear forms.) The pair $(\phi,\psi)\in\mathcal{P}_{\bf u}$ where the class of
${\bf u}$ is attained is unique up to a constant factor. OP with respect to a
($\mathbf{D}_{q}$-)semiclassical form of class $s$ are called a
($\mathcal{D}_{q}$-)semiclassical orthogonal polynomials of class $s$.

We end this section with the following definition.

######  Definition 2.3.

We call a pair of polynomials $(\phi,\psi)$,
$\phi(x)=a_{p}\,x^{p}+(\text{lower degree}$ $\text{terms})$ and
$\psi(x)=b_{q}\,x^{q}+(\text{lower degree terms})$
$(p\in\mathbb{N},q\in\mathbb{N}\setminus\\{0\\})$, admissible if $p-1\not=q$
or $a_{p}\,\gamma_{n}+b_{q}\,\alpha_{n-1}\not=0$ whenever $p-1=q$.

######  Remark 2.1.

We emphasize that from the point of view of [9], in this paper we are working
with the particular lattice $x(s)=(q^{-s}+q^{s})/2$. If we consider the
lattice $x(s)=\mathfrak{c}_{6}$, in the notation of [9], we have
$\alpha_{n-1}=1$ and $\gamma_{n}=n$, and so Definition 2.3 reduces the
admissibility condition for the standard derivative $($see [25, p. 119] and
[27, p. 46]$)$.

##  3\. Characterization theorems

The next theorem characterize $\mathcal{D}_{q}$-classical and
$\mathcal{D}_{q}$-semiclassical orthogonal polynomials from the structure
relation (4).

######  Theorem 3.1.

Let ${\bf u}\in\mathcal{P}^{\prime}$ be regular and let $(P_{n})_{n\geq 0}$
denote the corresponding sequence of orthogonal polynomials. The following
conditions are equivalent:

  * i)

There exist three nonzero polynomials, $\phi$, $\psi$ and $\rho$,
$(\psi,\rho)$ being an admissible pair, such that

$\displaystyle{\bf D}_{q}(\phi{\bf u})=\psi{\bf u},\qquad{\bf S}_{q}(\phi{\bf
u})=\rho{\bf u}.$

  * ii)

There exist $s\in\mathbb{N}$, complex numbers $(a_{n,j})_{j=0}^{n}$, and a
polynomial $\phi$ such that

(19)
$\displaystyle\phi\mathcal{D}_{q}P_{n}=\sum_{j=n-s}^{n+\deg\phi-1}a_{n,j}P_{j},$

with $a_{n,n-s}\neq 0$ for all $n\geq s$.

###### Proof.

$i)\implies ii)$: Write $\rho(x)=a_{r}\,x^{r}+(\text{lower degree}$
$\text{terms})$ and $\psi(x)=b_{s}\,x^{s}+(\text{lower degree terms})$
$(r\in\mathbb{N}\setminus\\{0\\},p\in\mathbb{N})$. Set
$d_{n}=a_{r}\alpha_{n-1}+b_{s}\gamma_{n}$. Clearly,

$\displaystyle\phi\mathcal{D}_{q}P_{n}=\sum_{j=0}^{n+\deg\phi-1}a_{n,j}P_{j},$

where

$\displaystyle a_{n,j}=\frac{\left\langle{\bf u},\phi
P_{j}\mathcal{D}_{q}P_{n}\right\rangle}{\left\langle{\bf
u},P_{j}^{2}\right\rangle}.$

From (13) we get

$\displaystyle\left\langle{\bf u},P_{j}^{2}\right\rangle a_{n,j}$
$\displaystyle=\left\langle{\bf u},\phi
P_{j}\mathcal{D}_{q}P_{n}\right\rangle=\left\langle\phi{\bf
u},P_{j}\mathcal{D}_{q}P_{n}\right\rangle$ $\displaystyle=\left\langle\phi{\bf
u},\mathcal{D}_{q}\left(\left(\mathcal{S}_{q}P_{j}-\alpha^{-1}\texttt{U}_{1}\mathcal{D}_{q}P_{j}\right)P_{n}\right)-\alpha^{-1}\mathcal{S}_{q}(P_{n}\mathcal{D}_{q}P_{j})\right\rangle$
$\displaystyle=-\left\langle{\bf D}_{q}(\phi{\bf
u}),\left(\mathcal{S}_{q}P_{j}-\alpha^{-1}\texttt{U}_{1}\mathcal{D}_{q}P_{j}\right)P_{n}\right\rangle-\alpha^{-1}\left\langle{\bf
S}_{q}(\phi{\bf u}),P_{n}\mathcal{D}_{q}P_{j}\right\rangle$
$\displaystyle=-\left\langle{\bf
u},\left(\psi\left(\mathcal{S}_{q}P_{j}-\alpha^{-1}\texttt{U}_{1}\mathcal{D}_{q}P_{j}\right)+\alpha^{-1}\rho\mathcal{D}_{q}P_{j}\right)P_{n}\right\rangle.$

There is no loss of generality in assuming $r-1\leq s$. For $r-1<s$, we have

$\displaystyle-\alpha\left\langle{\bf u},P_{j}^{2}\right\rangle
a_{n,j}=\left\\{\begin{array}[]{ll}a_{r}\alpha_{n-s-1}\left\langle{\bf
u},P_{n}^{2}\right\rangle,&j=n-s,\\\\[7.0pt] 0,&j<n-s.\end{array}\right.$

and for $r-1=s$, we get

$\displaystyle-\alpha\left\langle{\bf u},P_{j}^{2}\right\rangle
a_{n,j}=\left\\{\begin{array}[]{ll}d_{n-s}\left\langle{\bf
u},P_{n}^{2}\right\rangle,&j=n-s,\\\\[7.0pt] 0,&j<n-s.\end{array}\right.$

Hence $a_{n,n-s}\neq 0$, for $n\geq s$ and $ii)$ follows.

$ii)\implies i)$: Let $({\bf a}_{n})_{n\geq 0}$ be the dual basis associated
to $(P_{n})_{n\geq 0}$. Note that $ii)$ yields

$\displaystyle\left\langle{\bf D}_{q}(\phi{\bf a}_{n}),P_{j}\right\rangle$
$\displaystyle=-\left\langle{\bf
a}_{n},\phi\mathcal{D}_{q}P_{j}\right\rangle=-\sum_{l=j-s}^{j+\deg\phi-1}a_{j,l}\left\langle{\bf
a}_{n},P_{l}\right\rangle$
$\displaystyle=\left\\{\begin{array}[]{ll}-a_{j,n},&n-\deg\phi+1\leq j\leq
n+s,\\\\[7.0pt] 0,&\mbox{otherwise.}\end{array}\right.$

Writing

$\displaystyle{\bf D}_{q}(\phi{\bf a}_{n})=\sum_{j=0}^{\infty}\left\langle{\bf
D}_{q}(\phi{\bf a}_{n}),P_{j}\right\rangle{\bf a}_{j},$

in the sense of the weak dual topology in $\mathcal{P}^{\prime}$, and taking
into account that $\left\langle{\bf u},P_{n}^{2}\right\rangle{\bf
a}_{n}=P_{n}{\bf u}$, we get

$\displaystyle{\bf D}_{q}(\phi P_{n}{\bf u})=R_{n+s}{\bf u},\quad
R_{n+s}=-\left\langle{\bf
u},P_{n}^{2}\right\rangle\sum_{j=n-\deg\phi+1}^{n+s}\frac{a_{j,n}}{\left\langle{\bf
u},P_{j}^{2}\right\rangle}P_{j}.$

(Note that $R_{n+s}$ is a polynomial of degree $n+s$.) Taking $n=0$ and $n=1$
in the above expression, we have

(20) $\displaystyle{\bf D}_{q}(\phi{\bf u})$ $\displaystyle=R_{s}{\bf u},$
(21) $\displaystyle{\bf D}_{q}(\phi P_{1}{\bf u})$ $\displaystyle=R_{s+1}{\bf
u}.$

From (21), and using (14) and (20), we obtain

$\displaystyle\alpha R_{s+1}{\bf u}$ $\displaystyle=\alpha{\bf D}_{q}(\phi
P_{1}{\bf
u})=\big{(}\alpha\mathcal{S}_{q}P_{1}-\texttt{U}_{1}\mathcal{D}_{q}P_{1}\big{)}{\bf
D}_{q}(\phi{\bf u})+\mathcal{D}_{q}P_{1}{\bf S}_{q}(\phi{\bf u})$
$\displaystyle=(x-\alpha B_{0})R_{s}{\bf u}+{\bf S}_{q}(\phi{\bf u}).$

Hence

(22) $\displaystyle{\bf S}_{q}(\phi{\bf u})=\big{(}\alpha R_{s+1}-(x-\alpha
B_{0})R_{s}\big{)}{\bf u}.$

Note that $\alpha R_{s+1}-(x-\alpha B_{0})R_{s}\neq 0$. To obtain a
contradiction, suppose that the last assertion is false. Consequently,
$\phi{\bf u}=0$ with $\phi\neq 0$ and ${\bf u}$ regular, which is impossible.
We claim that $\big{(}R_{s},\alpha R_{s+1}-(x-\alpha B_{0})R_{s}\big{)}$ is an
admissible pair. According to Definition 2.3, this is equivalent to showing
that $d_{n}=a_{r}\alpha_{n-1}+b_{s}\gamma_{n}\neq 0$, where

$a_{r}=-\frac{\left\langle{\bf u},P_{0}^{2}\right\rangle}{\left\langle{\bf
u},P_{s}^{2}\right\rangle}a_{s,0},\quad b_{s}=-\alpha\frac{\left\langle{\bf
u},P_{1}^{2}\right\rangle}{\left\langle{\bf
u},P_{s+1}^{2}\right\rangle}a_{s+1,1}-a_{r}.$

For $\deg\,(\alpha R_{s+1}-(x-\alpha B_{0})R_{s})<s+1$, we have $b_{s}=0$, and
so $d_{n}=a_{r}\alpha_{n-1}\neq 0$. Assume $\deg\,(\alpha R_{s+1}-(x-\alpha
B_{0})R_{s})=s+1$. (Note that in this case $b_{s}\neq 0$.) We now claim

(23)
$\displaystyle\phi\mathcal{S}_{q}P_{n}=\displaystyle\sum_{j=n-s-1}^{n+\deg\phi}\widetilde{a}_{n,j}P_{j},\quad\widetilde{a}_{n,n-s-1}=-\alpha
a_{n,n-s}C_{n-s}+a_{n-1,n-s-1}C_{n},$

where we have assumed that $(P_{n})_{n\geq 0}$ satisfies (6). Indeed, apply
$\phi\mathcal{D}_{q}$ to (6) to obtain

$\displaystyle\phi(x)\mathcal{S}_{q}P_{n}(x)$
$\displaystyle=\phi(x)\left(-\alpha
x\mathcal{D}_{q}P_{n}(x)+\mathcal{D}_{q}P_{n+1}(x)+B_{n}\mathcal{D}_{q}P_{n}(x)+C_{n}\mathcal{D}_{q}P_{n-1}(x)\right)$
$\displaystyle=-\alpha
x\sum_{j=n-s}^{n+\deg\phi-1}a_{n,j}P_{j}(x)+\sum_{j=n-s+1}^{n+\deg\phi}a_{n+1,j}P_{j}(x)$
$\displaystyle\quad+B_{n}\sum_{j=n-s}^{n+\deg\phi-1}a_{n,j}P_{j}(x)+C_{n}\sum_{j=n-s-1}^{n+\deg\phi-2}a_{n-1,j}P_{j}(x),$

and (23) follows by using (6). We also claim that

(24) $\displaystyle
a_{n,n-s}=\left(k_{1}q^{n/2}+k_{2}q^{-n/2}\right)\prod_{j=n-s+1}^{n}C_{j},\qquad
n\geq s.$ (25) $\displaystyle
2\widetilde{a}_{n,n-s-1}=-(q^{1/2}-q^{-1/2})\left(k_{1}q^{n/2}-k_{2}q^{-n/2}\right)\prod_{j=n-s}^{n}C_{j},\qquad
n\geq s+1,$

where $k_{1}$ and $k_{2}$ are complex numbers. Indeed, apply
$\phi\mathcal{S}_{q}$ to (6) and use (12) to obtain

$\displaystyle\texttt{U}_{2}(x)\phi(x)\mathcal{D}_{q}P_{n}(x)+\alpha
x\phi(x)\mathcal{S}_{q}P_{n}(x)$
$\displaystyle\quad=\phi(x)\mathcal{S}_{q}P_{n+1}(x)+B_{n}\phi(x)\mathcal{S}_{q}P_{n}(x)+C_{n}\phi(x)\mathcal{S}_{q}P_{n-1}(x).$

Combining (19), (23) and (6), we obtain

$\sum_{j=n-s-2}^{n+\deg\phi+1}r_{n,j}P_{j}=0.$

Since $(P_{n})_{n\geq 0}$ is a free system, we have $r_{n,j}=0$ for all $j$.
By identifying the coefficient of $P_{n-s-2}$, we find

$\displaystyle
0=r_{n,n-s-2}=(\alpha^{2}-1)a_{n,n-s}C_{n-s}C_{n-s-1}+\alpha\widetilde{a}_{n,n-s-1}C_{n-s-1}-\widetilde{a}_{n-1,n-s-2}C_{n}.$

Using the expression of $\widetilde{a}_{n,n-s-1}$ given in (23), we get the
following second order linear homogeneous equation:

(26) $\displaystyle y(n)-2\alpha y(n-1)+y(n-2)=0,$

where

$y(n)=\displaystyle\frac{a_{n,n-s}}{\prod_{j=n-s+1}^{n}C_{j}},\qquad n\geq s.$

Note that $q^{1/2}$ and $q^{-1/2}$ are the solutions of the characteristic
equation of (26) and, therefore, we find

$\displaystyle y(n)=k_{1}q^{n/2}+k_{2}q^{-n/2},$

and (24) follows. Moreover, from the expression of $\widetilde{a}_{n,n-s-1}$
given in (23), (25) follows. Finally, using (24) and (25), we obtain

$\displaystyle d_{n}$ $\displaystyle=-\frac{\left\langle{\bf
u},P_{0}^{2}\right\rangle}{\left\langle{\bf
u},P_{s}^{2}\right\rangle}a_{s,0}\alpha_{n-1}+\frac{\left\langle{\bf
u},P_{0}^{2}\right\rangle}{\left\langle{\bf
u},P_{s+1}^{2}\right\rangle}\widetilde{a}_{s+1,0}\gamma_{n}$
$\displaystyle=\frac{\left\langle{\bf
u},P_{0}^{2}\right\rangle}{\left\langle{\bf
u},P_{s+1}^{2}\right\rangle}\left(-a_{s,0}C_{s+1}\alpha_{n-1}+\widetilde{a}_{s+1,0}\gamma_{n}\right)$
$\displaystyle=-\frac{1}{2}\left(2\alpha_{n-1}(k_{1}q^{s/2}+k_{2}q^{-s/2})+(q^{n/2}-q^{-n/2})(k_{1}q^{(s+1)/2}-k_{2}q^{-(s+1)/2})\right)$
$\displaystyle=-\alpha\big{(}k_{1}q^{(n+s)/2}+k_{2}q^{-(n+s)/2}\big{)}$
$\displaystyle=-\alpha\,a_{n+s,n}\prod_{j=1}^{n+s}C_{j}^{-1}\neq 0,$

and so $(R_{s},\alpha R_{s+1}-(x-\alpha B_{0})R_{s})$ is an admissible pair.
Thus, $i)$ follows from (20) and (22), and the theorem is proved. ∎

######  Remark 3.1.

A regular linear form ${\bf u}$ satisfying Theorem 3.1 $i)$ is classical or
semiclassical. Indeed, using (18) and (16) we get

$\displaystyle{\bf D}_{q}(\rho{\bf u})$ $\displaystyle={\bf D}_{q}{\bf
S}_{q}(\phi{\bf u})=(\alpha-\alpha^{-1}){\bf S}_{q}{\bf D}_{q}(\phi{\bf
u})+\alpha^{-1}\texttt{U}_{1}{\bf D}_{q}^{2}(\phi{\bf u})$
$\displaystyle=(\alpha-\alpha^{-1}){\bf S}_{q}(\psi{\bf
u})+\alpha^{-1}\texttt{U}_{1}{\bf D}_{q}(\psi{\bf u})$
$\displaystyle=(\alpha-\alpha^{-1}){\bf S}_{q}(\psi{\bf
u})+\alpha^{-1}(\alpha{\bf D}_{q}(\texttt{U}_{1}\psi{\bf
u})-(\alpha^{2}-1){\bf S}_{q}(\psi{\bf u}))$ $\displaystyle=\alpha{\bf
S}_{q}(\psi{\bf u})+{\bf D}_{q}(\texttt{U}_{1}\psi{\bf u}).$

Thus ${\bf D}_{q}\big{(}(\rho-\texttt{U}_{1}\psi){\bf u}\big{)}=\alpha{\bf
S}_{q}(\psi{\bf u})$ as claimed. Theorem 3.1 is the analogue of [21, Theorem
3.1] $($see also [25]$)$, from which a distributional version of [5, Theorem
1.1] follows.

Although Theorem 3.1 could be very useful in many situations it is little
precise regarding the classical or semiclassical character of the linear form.
In the following theorem we will be more precise in this sense, but in
counterpart we lose the direct connection with the equation (9).

######  Theorem 3.2.

Let ${\bf u}\in\mathcal{P}^{\prime}$ be regular and let $(P_{n})_{n\geq 0}$
denote the corresponding sequence of orthogonal polynomials satisfying (6).
Suppose that there exist a nonnegative integer $s$, complex numbers
$(a_{n,j})_{j=0}^{n}$, and a polynomial $\phi$ such that

(27)
$\displaystyle\phi\mathcal{D}_{q}P_{n}=\sum_{j=n-s}^{n+\deg\phi-1}a_{n,j}P_{j},$

with $a_{n.n-s}\neq 0$ for all $n\geq s$. Then there exist
$\Phi\in\mathcal{P}_{s+1}\setminus\mathcal{P}_{-1}$ and
$\Psi\in\mathcal{P}_{s}\setminus\mathcal{P}_{-1}$ such that

(28) $\displaystyle\Phi\,{\bf D}_{q}{\bf u}=\Psi\,{\bf S}_{q}{\bf u},$

where $\Phi$ and $\Psi$ never have more than $s-1$ common zeros. If $\Phi$ and
$\Psi$ have $s-1$ common zeros, then ${\bf u}$ is ${\bf D}_{q}$-classical.
Otherwise, ${\bf u}$ is ${\bf D}_{q}$-semiclassical of class $s-1-r$, $r$
being the number of common zeros of $\Phi$ and $\Psi$.

###### Proof.

We can now proceed analogously to the proof of Theorem 3.1 to obtain

$\displaystyle{\bf D}_{q}(\phi P_{n}{\bf u})=-Q_{n+s}{\bf u},\quad
Q_{n+s}=\left\langle{\bf
u},P_{n}^{2}\right\rangle\sum_{j=n-\deg\phi+1}^{n+s}\frac{a_{j,n}}{\left\langle{\bf
u},P_{j}^{2}\right\rangle}P_{j}.$

(Note that $Q_{n+s}$ is a polynomial of degree $n+s$.) Taking $n=0$ and $n=1$,
in the above expression, we have

(29) $\displaystyle{\bf D}_{q}(\phi{\bf u})$ $\displaystyle=-Q_{s}{\bf u},$
(30) $\displaystyle{\bf D}_{q}(\phi P_{1}{\bf u})$ $\displaystyle=-Q_{s+1}{\bf
u}.$

Using (29) and (14), we have

$\displaystyle-\alpha Q_{s+1}(x){\bf u}$ $\displaystyle=\alpha{\bf
D}_{q}\big{(}P_{1}(x)\phi{\bf u})=(x-\alpha B_{0}){\bf D}_{q}(\phi{\bf
u})+{\bf S}_{q}(\phi{\bf u})$ $\displaystyle=-(x-\alpha B_{0})Q_{s}(x){\bf
u}+{\bf S}_{q}(\phi{\bf u}),$

and so

$((x-\alpha B_{0})Q_{s}(x)-\alpha Q_{s+1}(x)){\bf u}={\bf S}_{q}(\phi{\bf
u}).$

Applying ${\bf D}_{q}$ to the above equation, and using (18), (29), and (16),
we can assert that

$\displaystyle-\alpha{\bf D}_{q}(((x-\alpha B_{0})Q_{s}(x)-\alpha
Q_{s+1}(x)){\bf u})$ $\displaystyle\quad=-\alpha{\bf D}_{q}{\bf
S}_{q}\big{(}\phi{\bf u}\big{)}=-(2\alpha^{2}-1){\bf S}_{q}{\bf
D}_{q}\big{(}\phi{\bf u}\big{)}-\texttt{U}_{1}{\bf D}_{q}^{2}\big{(}\phi{\bf
u}\big{)}$ $\displaystyle\quad=(2\alpha^{2}-1){\bf S}_{q}\big{(}Q_{s}(x){\bf
u}\big{)}+\texttt{U}_{1}{\bf D}_{q}\big{(}Q_{s}(x){\bf
u}\big{)}=\alpha^{2}{\bf S}_{q}\big{(}Q_{s}(x){\bf u}\big{)}+\alpha{\bf
D}_{q}\big{(}\texttt{U}_{1}Q_{s}(x){\bf u}\big{)}.$

Therefore,

(31) $\displaystyle{\bf D}_{q}\big{(}(Q_{s+1}(x)-(\alpha x-B_{0})Q_{s}(x)){\bf
u}\big{)}={\bf S}_{q}(Q_{s}(x){\bf u}),$

and $\mathbf{u}$ is classical or semiclassical of class at most $s-1$. Let us
now rewrite (31) in the form (28) to distinguish between cases. Using (14) and
(15), (31) becomes

$\displaystyle(\alpha\mathcal{S}_{q}R_{s+1}-\texttt{U}_{1}\mathcal{D}_{q}R_{s+1}+\big{(}\texttt{U}_{1}^{2}-\alpha^{2}\texttt{U}_{2}\big{)}\mathcal{D}_{q}Q_{s}){\bf
D}_{q}{\bf u}$
$\displaystyle\quad=\big{(}\texttt{U}_{1}\mathcal{D}_{q}Q_{s}+\alpha\mathcal{S}_{q}Q_{s}-\mathcal{D}_{q}R_{s+1}\big{)}{\bf
S}_{q}{\bf u},$

where $R_{s+1}(x)=Q_{s+1}(x)-(\alpha x-B_{0})Q_{s}(x)$. Consequently, (28)
follows with

(32) $\displaystyle\Phi$
$\displaystyle=\alpha\mathcal{S}_{q}R_{s+1}-\texttt{U}_{1}\mathcal{D}_{q}R_{s+1}+(\texttt{U}_{1}^{2}-\alpha^{2}\texttt{U}_{2})\mathcal{D}_{q}Q_{s},$
(33) $\displaystyle\Psi$
$\displaystyle=\texttt{U}_{1}\mathcal{D}_{q}Q_{s}+\alpha\mathcal{S}_{q}Q_{s}-\mathcal{D}_{q}R_{s+1}.$

Of course, $\Phi(x)\not=0$ and $\Psi(x)\not=0$, otherwise ${\bf u}={\bf 0}$,
which contradicts the regularity of $\mathbf{u}$. Without restriction of
generality, let us assume $\Phi(x)=(x-1)\Psi(x)$. From (28), and using (16)
and (17), we get

$\displaystyle{\bf D}_{q}\big{(}1/2(\alpha\,x-1){\bf u}\big{)}={\bf S}_{q}{\bf
u}.$

By Theorem 2.1, this leads to a contradiction with the regularity of
$\mathbf{u}$ —$a=d=0$ in the notation of Theorem 2.1 and so $d_{n}=0$
therein—, and the first part of the theorem follows. Now suppose that
$\Phi=\rho_{r}\phi$ and $\Psi=\rho_{r}\psi$, $r<s$ where
$\rho_{r}\in\mathcal{P}_{r}$, $\phi\in\mathcal{P}_{s-r+1}$ and
$\psi\in\mathcal{P}_{s-r}$. Hence (28) reduces to $\phi\,{\bf D}_{q}{\bf
u}=\psi\,{\bf S}_{q}{\bf u}$ and, therefore, using (16) and (17), we have

${\bf
D}_{q}\left((\mathcal{S}_{q}\phi+\texttt{U}_{2}\mathcal{D}_{q}\psi)\,{\bf
u}\right)={\bf S}_{q}\left((\mathcal{S}_{q}\psi+\mathcal{D}_{q}\phi)\,{\bf
u}\right).$

Since $\mathcal{S}_{q}\phi$ has degree at most $s-r+1$, and
$\mathcal{S}_{q}\psi$ and $\mathcal{D}_{q}\phi$ have degree at most $s-r$,
$\mathbf{u}$ is classical whenever $r=s-1$ or semiclassical of class at most
$s-r-1$ whenever $r<s-1$. Assume that (28) holds with $\Phi$ and $\Psi$ being
coprime, i.e., $r=0$. To obtain a contradiction, suppose that ${\bf u}$ is
semiclassical of class at most $s-2$: there exists $\phi\in\mathcal{P}_{s}$
and $\psi\in\mathcal{P}_{s-1}$ such that (9) holds. Taking into account (14)
and (15), (9) holds if and only if

(34) $\displaystyle\widetilde{\Phi}\,{\bf D}_{q}{\bf u}=\widetilde{\Psi}\,{\bf
S}_{q}{\bf u},$

where
$\widetilde{\Phi}=\alpha\mathcal{S}_{q}\phi-\texttt{U}_{1}\mathcal{D}_{q}\phi+(\texttt{U}_{1}^{2}-\alpha^{2}\texttt{U}_{2})\mathcal{D}_{q}\psi$
and
$\widetilde{\Psi}=\alpha\mathcal{S}_{q}\psi+\texttt{U}_{1}\mathcal{D}_{q}\psi-\mathcal{D}_{q}\phi$.
Combining (28) with (34) yields

$(\widetilde{\Psi}\,\Phi-\widetilde{\Phi}\,\Psi)\,{\bf D}_{q}{\bf
u}=\mathbf{0}.$

By the regularity of $\mathbf{u}$, and the fact that $\Phi$ and $\Psi$ are
coprime, $\Phi=a\widetilde{\Phi}$ and $\Psi=a\widetilde{\Psi}$
$(a\in\mathbb{C}\setminus\\{0\\})$, and so

$Q_{s}=a\,\psi,$

which is impossible. Thus ${\bf u}$ is semiclassical of class $s-1$. The same
conclusion can be drawn for $r\not=0$ and the theorem is proved.

∎

##  4\. Counterexamples

As an application of a particular case of Theorem 3.2, we disprove Conjecture
1.1 and Conjecture 1.2.

######  Proposition 4.1.

Let $(P_{n})_{n\geq 0}$ be a sequence of orthogonal polynomials satisfying (6)
with

$\displaystyle B_{n}=0,\qquad
C_{n}=\frac{1}{4}\big{(}1-(-1)^{n}q^{n/2}\big{)}\big{(}1-(-1)^{n}q^{(n-1)/2}\big{)}.$

Then $(P_{n})_{n\geq 0}$ is $\mathcal{D}_{q}$-semiclassical of class two and
the corresponding linear form satisfies (9) with

$\displaystyle\phi(x)$
$\displaystyle=-\frac{1}{8}(1-q^{-1})^{2}\big{(}4x^{4}-(q+5)x^{2}+q+1),$
$\displaystyle\psi(x)$ $\displaystyle=\frac{1}{4}(q-1)q^{-3/2}x(4x^{2}-3-q).$

###### Proof.

We claim that $(P_{n})_{n\geq 0}$ satisfies

(35) $\displaystyle\mathcal{S}_{q}P_{n}$
$\displaystyle=\alpha_{n}P_{n}+b_{n}C_{n-1}P_{n-2},$ (36)
$\displaystyle\texttt{U}_{2}\mathcal{D}_{q}P_{n}$
$\displaystyle=a_{n}P_{n+1}+c_{n}P_{n-1}+d_{n}P_{n-3},$

with

$\displaystyle a_{n}$ $\displaystyle=(\alpha^{2}-1)\gamma_{n},$ $\displaystyle
b_{n}$
$\displaystyle=-\frac{1}{2}\big{(}1-(-1)^{n}q^{n/2}\big{)}\big{(}(-1)^{n}-q^{-(n-1)/2}\big{)},$
$\displaystyle c_{n}$ $\displaystyle=b_{n+1}C_{n}-\alpha
b_{n}C_{n-1}-(\alpha^{2}-1)\gamma_{n}C_{n},\qquad d_{n}=(b_{n-1}C_{n}-\alpha
b_{n}C_{n-1})C_{n-2}.$

Indeed, we prove this by induction on $n$. For $n=1$, RHS of (35) gives
$\alpha_{1}P_{1}(x)+b_{1}C_{0}P_{-1}(x)=\alpha x,$ while LHS gives
$\mathcal{S}_{q}P_{1}(x)=\mathcal{S}_{q}x=\alpha x$. Similarly, for $n=1$, LHS
of (36) gives $\texttt{U}_{2}\mathcal{D}_{q}P_{1}=\texttt{U}_{2}$, while RHS
gives

$\displaystyle a_{1}P_{2}(x)+c_{1}P_{0}(x)+d_{1}P_{-1}(x)$
$\displaystyle=a_{1}P_{2}(x)+c_{1}$
$\displaystyle=(\alpha^{2}-1)(x^{2}-C_{1})+b_{2}C_{1}-\alpha
b_{1}C_{0}-(\alpha^{2}-1)C_{1}$ $\displaystyle=(\alpha^{2}-1)(x^{2}-1).$

Assuming (35) and (36) hold, with $k$ instead of $n$, for $k=1,2,\dots,n$, we
will prove it for $k=n+1$. Apply $\mathcal{S}_{q}$ to (6), and use (12), to
obtain

$\displaystyle\mathcal{S}_{q}(P_{n+1}(x)+C_{n}P_{n-1}(x))$
$\displaystyle=\mathcal{S}_{q}(xP_{n}(x))=\texttt{U}_{2}(x)\mathcal{D}_{q}P_{n}(x)+\alpha
x\mathcal{S}_{q}P_{n}(x).$

From (36) for $n$ and (35) for $n-1$ and $n$, we get

(37) $\displaystyle\mathcal{S}_{q}P_{n+1}(x)$
$\displaystyle=a_{n}P_{n+1}(x)+c_{n}P_{n-1}(x)+d_{n}P_{n-3}(x)$
$\displaystyle\quad+\alpha x(\alpha_{n}P_{n}(x)+b_{n}C_{n-1}P_{n-2}(x))$
$\displaystyle\quad-C_{n}(\alpha_{n-1}P_{n-1}(x)+b_{n-1}C_{n-2}P_{n-3}(x)).$

Now using (6), (37) becomes

$\displaystyle\mathcal{S}_{q}P_{n+1}$
$\displaystyle=(a_{n}+\alpha\alpha_{n})P_{n+1}$
$\displaystyle+(c_{n}+\alpha\alpha_{n}C_{n}+\alpha
b_{n}C_{n-1}-\alpha_{n-1}C_{n})P_{n-1}$ $\displaystyle+(d_{n}+\alpha
b_{n}C_{n-1}C_{n-2}-b_{n-1}C_{n}C_{n-2})P_{n-3}.$

The reader should convince himself that the following relations hold:

$\displaystyle\alpha_{n+1}$ $\displaystyle=a_{n}+\alpha\alpha_{n},$
$\displaystyle b_{n+1}C_{n+1}$
$\displaystyle=c_{n}+\alpha\alpha_{n}C_{n}+\alpha
b_{n}C_{n-1}-\alpha_{n-1}C_{n},$ $\displaystyle 0$ $\displaystyle=d_{n}+\alpha
b_{n}C_{n-1}C_{n-2}-b_{n-1}C_{n}C_{n-2}.$

This gives $\mathcal{S}_{q}P_{n+1}=\alpha_{n+1}P_{n+1}+b_{n+1}C_{n}P_{n-1}$,
and (35) holds for $n+1$. Similarly, apply $\texttt{U}_{2}\mathcal{D}_{q}$ to
(6), and use (11), to obtain

$\displaystyle\texttt{U}_{2}(x)\mathcal{D}_{q}(P_{n+1}(x)+C_{n}P_{n-1}(x))=\texttt{U}_{2}(x)\mathcal{D}_{q}(xP_{n}(x))=\texttt{U}_{2}(x)(\mathcal{S}_{q}P_{n}(x)+\alpha
x\mathcal{D}_{q}P_{n}(x))$

or, using (35) for $n$,

(38) $\displaystyle\texttt{U}_{2}(x)\mathcal{D}_{q}P_{n+1}(x)$
$\displaystyle=\texttt{U}_{2}(x)\mathcal{S}_{q}P_{n}(x)+\alpha
x\texttt{U}_{2}(x)\mathcal{D}_{q}P_{n}(x)-C_{n}\texttt{U}_{2}(x)\mathcal{D}_{q}P_{n-1}(x)$
$\displaystyle=\texttt{U}_{2}(x)\big{(}\alpha_{n}P_{n}(x)+b_{n}C_{n-1}P_{n-2}(x)\big{)}+\alpha
x\texttt{U}_{2}(x)\mathcal{D}_{q}P_{n}(x)$ $\displaystyle-
C_{n}\texttt{U}_{2}(x)\mathcal{D}_{q}P_{n-1}(x).$

From (6) it follows that

$\displaystyle\texttt{U}_{2}P_{n}=(\alpha^{2}-1)(P_{n+2}+(C_{n+1}+C_{n}-1)P_{n}+C_{n}C_{n-1}P_{n-2}).$

Combining the above equation with (36) for $n-1$ and $n$, (38) becomes

$\displaystyle\texttt{U}_{2}\mathcal{D}_{q}P_{n+1}$
$\displaystyle=\big{(}(\alpha^{2}-1)\alpha_{n}+\alpha a_{n}\big{)}P_{n+2}$
$\displaystyle\quad+\big{(}(\alpha^{2}-1)\alpha_{n}(C_{n}+C_{n+1}-1)+(\alpha^{2}-1)b_{n}C_{n-1}+\alpha
a_{n}C_{n+1}$ $\displaystyle\quad\quad+\alpha c_{n}-a_{n-1}C_{n}\big{)}P_{n}$
$\displaystyle\quad+\big{(}(\alpha^{2}-1)\alpha_{n}C_{n}C_{n-1}+(\alpha^{2}-1)b_{n}C_{n-1}(C_{n-1}+C_{n-2}-1)$
$\displaystyle\quad\quad+\alpha c_{n}C_{n-1}+\alpha
d_{n}-c_{n-1}C_{n}\big{)}P_{n-2}$ $\displaystyle\quad+\big{(}\alpha
d_{n}C_{n-3}-d_{n-1}C_{n}+(\alpha^{2}-1)b_{n}C_{n-1}C_{n-2}C_{n-3}\big{)}P_{n-4}.$

The reader again should convince himself that the following relations hold:

$\displaystyle a_{n+1}$ $\displaystyle=(\alpha^{2}-1)\alpha_{n}+\alpha a_{n},$
$\displaystyle c_{n+1}$
$\displaystyle=(\alpha^{2}-1)\alpha_{n}(C_{n}+C_{n+1}-1)+(\alpha^{2}-1)b_{n}C_{n-1}+\alpha
a_{n}C_{n+1}$ $\displaystyle\quad+\alpha c_{n}-a_{n-1}C_{n},$ $\displaystyle
d_{n+1}$
$\displaystyle=(\alpha^{2}-1)\alpha_{n}C_{n}C_{n-1}+(\alpha^{2}-1)b_{n}C_{n-1}(C_{n-1}+C_{n-2}-1)$
$\displaystyle\quad+\alpha c_{n}C_{n-1}+\alpha d_{n}-c_{n-1}C_{n},$
$\displaystyle 0$ $\displaystyle=\alpha
d_{n}C_{n-3}-d_{n-1}C_{n}+(\alpha^{2}-1)b_{n}C_{n-1}C_{n-2}C_{n-3}.$

We thus get

$\texttt{U}_{2}\mathcal{D}_{q}P_{n+1}=a_{n+1}P_{n+2}+c_{n+1}P_{n}+d_{n+1}P_{n-2},$

as claimed. Observe from (36) that $(P_{n})_{n\geq 0}$ satisfies the
hypotheses of Theorem 3.2 with $B_{n}=0$ and $\phi=\texttt{U}_{2}$. Note also
that

$\displaystyle C_{1}$ $\displaystyle=\frac{1}{2}(1+q^{1/2}),\qquad$
$\displaystyle C_{2}$ $\displaystyle=\frac{1}{4}(1-q)(1-q^{1/2}),\qquad$
$\displaystyle C_{3}$ $\displaystyle=\frac{1}{4}(1+q)(1+q^{3/2}),\qquad$
$\displaystyle C_{4}$ $\displaystyle=\frac{1}{4}(1-q^{2})(1-q^{3/2}).$

Under the notation of Theorem 3.2 and its proof, we get

(39) $\displaystyle Q_{3}(x)$
$\displaystyle=\frac{c_{1}}{C_{1}}P_{1}(x)+\frac{d_{3}}{C_{3}C_{2}C_{1}}P_{3}(x)=\frac{1}{4}(q-1)q^{-3/2}x(4x^{2}-3-q),$
$\displaystyle Q_{4}(x)$
$\displaystyle=\frac{c_{2}}{C_{2}}P_{2}(x)+\frac{d_{4}}{C_{4}C_{3}C_{2}}P_{4}(x)=\frac{1}{8}(q-1)q^{-2}(8x^{4}-8x^{2}+1-q^{2}),$
(40) $\displaystyle R_{4}(x)$ $\displaystyle=Q_{4}(x)-\alpha
xQ_{3}(x)=-\frac{1}{8}(1-q^{-1})^{2}(4x^{4}-(q+5)x^{2}+q+1).$

Taking into account that $\mathcal{S}_{q}x=\alpha x$,
$\mathcal{D}_{q}x^{2}=2\alpha x$,

$\displaystyle\mathcal{S}_{q}x^{2}$
$\displaystyle=(2\alpha^{2}-1)x^{2}+1-\alpha^{2},$
$\displaystyle\mathcal{D}_{q}x^{3}$
$\displaystyle=(4\alpha^{2}-1)x^{2}+1-\alpha^{2},\quad\mathcal{S}_{q}x^{3}=\alpha(4\alpha^{2}-3)x^{3}+3\alpha(1-\alpha^{2})x,$
$\displaystyle\mathcal{D}_{q}x^{4}$
$\displaystyle=4\alpha(2\alpha^{2}-1)x^{3}+4\alpha(1-\alpha^{2})x,$
$\displaystyle\mathcal{S}_{q}x^{4}$
$\displaystyle=(8\alpha^{4}-8\alpha^{2}+1)x^{4}+2(1-\alpha^{2})(4\alpha^{2}-1)x^{2}+(1-\alpha^{2})^{2},$

from (32) and (33), we have

$\displaystyle\Phi(x)$
$\displaystyle=-\frac{1}{16}(q-1)^{2}q^{-3/2}(8qx^{4}-2(q^{2}+4q+1)x^{2}+(q+1)^{2}),$
$\displaystyle\Psi(x)$
$\displaystyle=\frac{1}{4}(q^{1/2}-q^{-1/2})(4qx^{2}-3q-1)x.$

Finally, since $\Phi$ and $\Psi$ are coprime, by Theorem 3.2, (31), (39) and
(40), the result follows. ∎

######  Remark 4.1.

The structure relation (36) is of type (4) with $\phi=\texttt{U}_{2}$, $M=3$,
and $N=1$. Consequently, the semiclassical orthogonal polynomials given in
Proposition 4.1 disprove Conjecture 1.1.

######  Corollary 4.1.

Assume the hypotheses and notation of Proposition 4.1. Then $(P_{n})_{n\geq
0}$ satisfies

(41)
$\displaystyle(\alpha^{2}-1)^{2}(x^{2}-\alpha^{2})(1-x^{2})\mathcal{D}_{q}^{2}P_{n}(x)$
$\displaystyle=-(\alpha^{2}-1)^{2}\gamma_{n}\gamma_{n-1}P_{n+2}(x)+d_{n,1}P_{n}(x)$
$\displaystyle+d_{n,2}P_{n-2}(x)+d_{n,3}P_{n-4}(x)+d_{n,4}P_{n-6}(x),$

with

$\displaystyle d_{n,1}$
$\displaystyle=a_{n}c_{n+1}+a_{n-1}c_{n}-2\alpha(\alpha^{2}-1)(\alpha_{n}^{2}-1)(C_{n+1}+C_{n}-1)$
$\displaystyle\quad-4\alpha^{2}(\alpha^{2}-1)\alpha_{n-1}b_{n}C_{n-1},$
$\displaystyle d_{n,2}$
$\displaystyle=a_{n}d_{n+1}+c_{n}c_{n-1}+a_{n-3}d_{n}-2\alpha(\alpha^{2}-1)(\alpha_{n}^{2}-1)C_{n}C_{n-1}$
$\displaystyle\quad-4\alpha^{2}(\alpha^{2}-1)\alpha_{n-1}b_{n}C_{n-1}(C_{n-1}+C_{n-2}-1)-2\alpha(\alpha^{2}-1)b_{n}b_{n-2}C_{n-1}C_{n-3},$
$\displaystyle d_{n,3}$
$\displaystyle=c_{n}d_{n-1}+c_{n-3}d_{n}-4\alpha^{2}(\alpha^{2}-1)\alpha_{n-1}b_{n}C_{n-1}C_{n-2}C_{n-3}$
$\displaystyle\quad-2\alpha(\alpha^{2}-1)b_{n}b_{n-2}C_{n-1}C_{n-3}(C_{n-3}+C_{n-4}-1),$
$\displaystyle d_{n,4}$
$\displaystyle=-4\alpha^{2}q^{-(n-3)/2}C_{n}C_{n-1}C_{n-2}C_{n-3}C_{n-4}C_{n-5}.$

###### Proof.

From the previous result, we apply $\texttt{U}_{2}\mathcal{D}_{q}$ to (36),
and use (11), to get

$\displaystyle\texttt{U}_{2}\mathcal{S}_{q}\texttt{U}_{2}\mathcal{D}_{q}^{2}P_{n}+\texttt{U}_{2}\mathcal{D}_{q}\texttt{U}_{2}\mathcal{S}_{q}\mathcal{D}_{q}P_{n}=\texttt{U}_{2}\mathcal{D}_{q}(a_{n}P_{n+1}+c_{n}P_{n-1}+d_{n}P_{n-3}),$

and since
$\mathcal{S}_{q}\texttt{U}_{2}=\alpha^{2}\texttt{U}_{2}+\texttt{U}_{1}^{2}$
and $\mathcal{D}_{q}\texttt{U}_{2}=2\alpha\texttt{U}_{1}$, we use again (36)
to obtain

(42) $\displaystyle(\alpha^{2}\texttt{U}_{2}$
$\displaystyle+\texttt{U}_{1}^{2})\texttt{U}_{2}\mathcal{D}_{q}^{2}P_{n}+2\alpha\texttt{U}_{1}\texttt{U}_{2}\mathcal{S}_{q}\mathcal{D}_{q}P_{n}=a_{n}a_{n+1}P_{n+2}$
$\displaystyle+(a_{n}c_{n+1}+a_{n-1}c_{n})P_{n}+(a_{n}d_{n+1}+c_{n}c_{n-1}+a_{n-3}d_{n})P_{n-2}$
$\displaystyle+(c_{n}d_{n-1}+d_{n}c_{n-3})P_{n-4}+d_{n}d_{n-3}P_{n-6}.$

On the other hand, it is known from [9, Lemma 2.1] that

$\displaystyle\alpha\mathcal{S}_{q}^{2}P_{n}$
$\displaystyle=\mathcal{S}_{q}(\texttt{U}_{1}\mathcal{D}_{q}P_{n})+\texttt{U}_{2}\mathcal{D}_{q}^{2}P_{n}+\alpha
P_{n}=\alpha^{2}\textbf{U}_{2}\mathcal{D}_{q}^{2}P_{n}+\alpha\texttt{U}_{1}\mathcal{S}_{q}\mathcal{D}_{q}P_{n}+\alpha
P_{n},$

where the second equality holds thanks to (12). Now we apply $\mathcal{S}_{q}$
to (35) using again the same equation and the above equation in order to
obtain

(43)
$\displaystyle\alpha\texttt{U}_{2}(x)\mathcal{D}_{q}^{2}P_{n}(x)+\texttt{U}_{1}(x)\mathcal{S}_{q}\mathcal{D}_{q}P_{n}(x)$
$\displaystyle=(\alpha_{n}^{2}-1)P_{n}(x)+2\alpha\alpha_{n-1}b_{n}C_{n-1}P_{n-2}(x)$
$\displaystyle+b_{n}b_{n-2}C_{n-1}C_{n-3}P_{n-4}(x).$

Therefore, (41) holds by combining (42) with (43) in order to eliminate
$\mathcal{S}_{q}\mathcal{D}_{q}P_{n}$ and by using (6), with

$\displaystyle
d_{n,4}=d_{n}d_{n-3}-2\alpha(\alpha^{2}-1)b_{n}b_{n-2}C_{n-1}C_{n-3}C_{n-4}C_{n-5}.$

In addition, $b_{n}=2C_{n}q^{-(n-1)/2}$ and
$d_{n}=(q-1)q^{-n/2}C_{n}C_{n-1}C_{n-2}$. Therefore we obtain

$\displaystyle d_{n,4}=-4\alpha^{2}q^{-(n-3)/2}\prod_{j=0}^{5}C_{n-j}\neq 0.$

The result is then proved. ∎

######  Remark 4.2.

The structure relation (41) is of type (4) with
$\phi(x)=(\alpha^{2}-1)^{2}(x^{2}-\alpha^{2})(1-x^{2})$, $M=6$, and $N=2$.
Consequently, the semiclassical orthogonal polynomials given in Proposition
4.1 disprove Conjecture 1.2.

##  5\. Further results: $D_{q,\omega}$-semiclassical orthogonal polynomials

Although this paper was originally intended to deal with the Askey-Wilson
operator, the ideas developed above allow working with other operators. The
results of this section are related to the structure relation that appears in
[14, Conjecture 24.7.7], and warn the reader of the existence of semiclassical
OP in such a problem. Recall that given complex numbers $q$ and $\omega$,
Hahn’s operator $D_{q,\omega}:\mathcal{P}\to\mathcal{P}$ is defined by

$\displaystyle D_{q,\omega}f(x):=\frac{f(qx+\omega)-f(x)}{(q-1)x+\omega},$

where we have fixed $q$ and $\omega$ such that

(44) $\displaystyle|1-q|+|\omega|\not=0,\qquad
q\not\in\\{0\\}\cup\left\\{e^{2ij\pi/n}\left|\right.1\leq j\leq
n-1,n\in\mathbb{N}\setminus\\{0,1\\}\right\\}.$

For every ${\bf u}\in\mathcal{P^{*}}$ and $f\in\mathcal{P}$, $D_{q,\omega}$
induces ${\bf D}_{q,\omega}:\mathcal{P}^{*}\to\mathcal{P}^{*}$ defined by

$\displaystyle\langle{\bf D}_{q,\omega}{\bf u},f\rangle=-q^{-1}\langle{\bf
u},D_{q,\omega}^{*}f\rangle,$

where $D_{q,\omega}^{*}=D_{1/q,-\omega/q}$.

######  Definition 5.1.

[4, p. 487] ${\bf u}\in\mathcal{P}^{*}$ is called ${\bf
D}_{q,\omega}$-classical if it is regular and there exist
$\phi\in\mathcal{P}_{2}\setminus\mathcal{P}_{-1}$ and
$\psi\in\mathcal{P}_{1}\setminus\mathcal{P}_{-1}$ such that

(45) $\displaystyle{\bf
D}_{q,\omega}\left(\phi\,\mathbf{u}\right)=\psi\,\mathbf{u}.$

$($We will call it simply classical when no confusion can arise.$)$

######  Definition 5.2.

[4, p. 855] We call a linear form, in $\mathcal{P}^{*}$, ${\bf
D}_{q,\omega}$-semiclassical if it is regular, not ${\bf
D}_{q,\omega}$-classical, and there exist two polynomials $\phi$ and $\psi$
with at least one of them nonzero, such that (45) holds. $($We will call it
simply semiclassical when no confusion can arise.$)$

OP with respect to a ($\mathbf{D}_{q,\omega}$-)semiclassical form of class $s$
is called ($D_{q,\omega}$-) semiclassical of class $s$. Under the conditions
of Definition 5.2, we define the class of ${\bf u}$ as in Section 2. The next
theorem is the analogue of Theorem 2.1. Here we use the standard notation

$\displaystyle[n]_{q}=\frac{q^{n}-1}{q-1}.$

######  Theorem 5.1.

[3, Theorem 1.2] Suppose that ${\bf u}\in\mathcal{P}^{*}$ satisfies (45) with
$\phi(x)=ax^{2}+bx+c$ and $\psi(x)=dx+e$. Then ${\bf u}$ is regular if and
only if

$\displaystyle d_{n}\neq 0,\qquad\phi\left(-\frac{e_{n}}{d_{2n}}\right)\neq
0,$

for all $n\in\mathbb{N}$, where $d_{n}=d\,q^{n}+a[n]_{q}$ and
$e_{n}=eq^{n}+(\omega d_{n}+b)[n]_{q}$. Moreover, $(P_{n})_{n\geq 0}$
satisfies (6) with

$\displaystyle
B_{n}=\omega[n]_{q}+\frac{[n]_{q}e_{n-1}}{d_{2n-2}}-\frac{[n+1]_{q}e_{n}}{d_{2n}},$
$\displaystyle
C_{n+1}=-\frac{q^{n}[n+1]_{q}d_{n-1}}{d_{2n-1}d_{2n+1}}\,\phi\left(-\frac{e_{n}}{d_{2n}}\right).$

In this context we have also an analogue to Theorem 3.2 for semiclassical
orthogonal polynomials of class one.

######  Theorem 5.2.

Let ${\bf u}\in\mathcal{P}^{\prime}$ be regular and let $(P_{n})_{n\geq 0}$
denote the corresponding sequence of orthogonal polynomials satisfying (6).
Suppose that there exist complex numbers $c$, $(a_{n})_{n\geq 0}$,
$(b_{n})_{n\geq 0}$ $($$b_{n}\not=0$$)$, and $(c_{n})_{n\geq 0}$ such that

(46)
$\displaystyle(x-c)D_{q,\omega}P_{n}(x)=a_{n}P_{n}(x)+(b_{n}x+c_{n})P_{n-1}(x).$

Then

$\displaystyle{\bf D}_{1/q,-\omega/q}\big{(}(x-c){\bf
u}\big{)}=-\frac{qb_{2}}{C_{2}}(x-\lambda_{+})(x-\lambda_{-}){\bf u},$

where

$\displaystyle\lambda_{\pm}=\frac{1}{2}(B_{0}+B_{1})+\frac{c-B_{0}}{2b_{2}C_{1}}C_{2}\pm\frac{1}{2}\left((B_{0}-B_{1}-\frac{c-B_{0}}{b_{2}C_{1}}C_{2})^{2}+4C_{1}\right)^{1/2}.$

If $q(\omega+qc-\lambda_{+})(\omega+qc-\lambda_{-})=-C_{2}/b_{2}$, then ${\bf
u}$ is ${\bf D}_{1/q,-\omega/q}$-classical. More precisely, $(P_{n})_{n\geq
0}$ are the Al-Salam-Carlitz polynomials. Otherwise, ${\bf u}$ is a ${\bf
D}_{1/q,-\omega/q}$-semiclassical of class one. Moreover, if

(47) $\displaystyle
b_{2}C_{1}^{2}=(B_{0}-c)(b_{2}(B_{1}-c)C_{1}-(B_{0}-c)C_{2}),$

then

$\displaystyle{\bf u}=(x-c)^{-1}{\bf v}+\delta_{c},$

${\bf v}$ being the linear form corresponding to the Al-Salam-Carlitz
polynomials.

###### Proof.

As in the proof of Theorem 3.2, from (6) and (46) we get

$\displaystyle{\bf D}_{1/q,-\omega/q}\big{(}(x-c){\bf
a}_{n}\big{)}=-q(a_{n}+b_{n}){\bf a}_{n}-q(c_{n+1}+b_{n+1}B_{n}){\bf
a}_{n+1}-q\,b_{n+2}C_{n+1}{\bf a}_{n+2},$

in the sense of the weak dual topology in $\mathcal{P}^{\prime}$, $({\bf
a}_{n})_{n\geq 0}$ being the dual basis associated to $(P_{n})_{n\geq 0}$.
Taking $n=0$ in the above expression we have

(48) $\displaystyle{\bf D}_{1/q,-\omega/q}\big{(}(x-c){\bf
u}\big{)}=\varphi(x){\bf u},$

where $\varphi$ is a polynomial of degree two given by

$\displaystyle\varphi=-\frac{q}{C_{1}C_{2}}((a_{0}+b_{0})C_{1}C_{2}+(r_{1}+b_{1}B_{0})C_{2}P_{1}+b_{2}C_{1}P_{2}).$

Taking $n=0$, $n=1$, and $n=2$ in (46) we get

$\displaystyle a_{0}+b_{0}=0,\qquad c_{1}+b_{1}B_{0}=B_{0}-c,\qquad
b_{2}=q+1+\frac{(B_{0}-c)(qB_{0}-B_{1}+\omega)}{C_{1}}.$

Hence $C_{2}\varphi(x)=-q\,b_{2}(x-\lambda_{+})(x-\lambda_{-})$, and the first
part of the theorem follows. Assume that $\varphi(\omega+q\,c)=1$. Recall that
(see [3, (2.10)])

$\displaystyle{\bf D}_{1/q,-\omega/q}(f{\bf u})=D_{1/q,-\omega/q}f\,{\bf
u}+f\left((x-\omega)/q\right){\bf D}_{1/q,-\omega/q}{\bf
u}\qquad(f\in\mathcal{P}).$

Using this identity, (48) becomes

$\displaystyle(x-qc-\omega){\bf D}_{1/q,-\omega/q}{\bf u}=q(\varphi(x)-1){\bf
u}.$

Since $\varphi(\omega+qc)=1$, $x-qc-\omega$ and $q(\varphi(x)-1)$ have a
common zero at $x=qc+\omega$, and therefore there exists a polynomial of
degree one, $Q_{1}$, such that $q(\varphi(x)-1)=(x-qc-\omega)Q_{1}(x)$, which
gives

$\displaystyle(x-qc-\omega)\Big{(}({\bf D}_{1/q,-\omega/q}{\bf
u})-Q_{1}(x){\bf u}\Big{)}=0,$

and so

$\displaystyle{\bf D}_{1/q,-\omega/q}{\bf
u}=\frac{1}{(q^{-1}-1)rs}(x-r-s-\omega/(1-q)){\bf u},$

for some nonzero complex numbers $r$ and $s$, i.e.
$Q_{1}(x)=1/((q^{-1}-1){rs})(x-r-s-\omega/(1-q))$. This last equation is of
type (45) with $\phi=1$ and $\psi(x)=1/((q^{-1}-1){rs})(x-r-s-\omega/(1-q))$.
We claim that $\mathbf{u}$ is regular. Indeed, by Theorem 5.1, $c=1$, $b=0$,
$a=0$, $d=1/((q^{-1}-1)rs)$, and $e=-(r+s+\omega/(1-q))/((q^{-1}-1)rs)$. Hence

$d_{n}=q^{-n}/((q^{-1}-1)rs)\neq 0,\qquad\phi=1\neq 0.$

Moreover, by Theorem 5.1, we get

$\displaystyle B_{n}$
$\displaystyle=-q^{-1}\omega[n]_{q^{-1}}+\frac{[n]_{q^{-1}}e_{n-1}}{d_{2n-2}}-\frac{[n+1]_{q^{-1}}e_{n}}{d_{2n}}$
$\displaystyle=\omega/(1-q)+(r+s)q^{n},$ $\displaystyle C_{n+1}$
$\displaystyle=-\frac{q^{-n}[n+1]_{q^{-1}}d_{n-1}}{d_{2n-1}d_{2n+1}}=-rs(1-q^{n+1})q^{n},$

and finally $P_{n}(x)=s^{n}U_{n}^{(r/s)}((x-\omega/(1-q))/s\,|\,q)$, where
$(U_{n}^{(a)}(\cdot\,|\,q))_{n\geq 0}$ are the Al-Salam-Carlitz polynomial
(see [17, Section 14.24]). Assume now that $\varphi(\omega+qc)\neq 1$. Hence
$x-\omega-qc$ and $\varphi(x)-1$ are coprime. Using the same argumentum ad
absurdum as in Theorem 3.2, we see that ${\bf u}$ is a ${\bf
D}_{1/q,-\omega/q}$-semiclassical form of class one. Now, from (48), and using
(47), we get

$\lambda_{+}=c,\qquad\lambda_{-}=c-\left(\left(B_{0}-B_{1}-\frac{c-B_{0}}{b_{2}C_{1}}C_{2}\right)^{2}+4C_{1}\right)^{1/2}.$

Then (48) becomes

$\displaystyle{\bf D}_{1/q,-\omega/q}\big{(}(x-c){\bf
u}\big{)}=-\frac{qb_{2}}{C_{2}}(x-c)(x-c+\Delta^{1/2}){\bf u},$

where
$\Delta=\left(B_{0}-B_{1}-\displaystyle\frac{c-B_{0}}{b_{2}C_{1}}C_{2}\right)^{2}+4C_{1}$
or, equivalently,

$\displaystyle{\bf D}_{1/q,-\omega/q}((x-c){\bf
u})=\frac{1}{(q^{-1}-1)rs}(x-c)(x-r-s-\omega/(1-q)){\bf u},$

where $r$ and $s$ are nonzero complex numbers such that $(q-1)rs=C_{2}/b_{2}$
and $r+s=c-\omega/(1-q)-\Delta^{1/2}$. Define ${\bf v}=(x-c){\bf u}$. Hence

$\displaystyle{\bf D}_{1/q,-\omega/q}{\bf
v}=\frac{1}{(q^{-1}-1)rs}(x-r-s-\omega/(1-q)){\bf v}.$

As above, by Theorem 5.1,

$d_{n}=\frac{q^{-n}}{(q^{-1}-1)rs}\neq 0,\qquad\phi=1\neq 0.$

Moreover, also by Theorem 5.1, ${\bf v}$ is the linear form corresponding to
the Al-Salam-Carlitz polynomials, and the theorem follows. ∎

The next proposition gives an explicit example of semiclassical orthogonal
polynomials of class one satisfying (46), which prevents the reader from
making any conjectures related to classical polynomials when faced with a
relation of type (1) after changing the standard derivative by Hahn’s
operator.

######  Proposition 5.1.

Fix $\omega,q\in\mathbb{C}$ such that (44) hold. Fix $a,b\in\mathbb{C}$ such
that

$-a\not=b,\qquad b\not=0,\qquad a+(-1)^{n}b-(a+b)q^{n}\neq 0,$

for all $n\in\mathbb{N}$. Let $(P_{n})_{n\geq 0}$ be a sequence of orthogonal
polynomials satisfying (6) with

(49) $\displaystyle B_{n}=\frac{\omega}{1-q},\qquad
C_{n}=(a+b)\left(\frac{a+(-1)^{n}b}{a+b}-q^{n}\right)q^{n}.$

Then $(P_{n})_{n\geq 0}$ is a $D_{1/q,-\omega/q}$-semiclassical of class one
and its corresponding linear form $\mathbf{u}\in\mathcal{P}^{\prime}$
satisfies

$\displaystyle{\bf
D}_{1/q,-\omega/q}\left(\left(x-\frac{\omega}{1-q}\right)\mathbf{u}\right)$
$\displaystyle\quad=\frac{1}{(a+b)(q-1)}\left(\frac{1}{q}\left(x-\frac{\omega}{1-q}\right)^{2}+b-a+(a+b)q\right){\bf
u}.$

###### Proof.

We claim that $(P_{n})_{n\geq 0}$ satisfies (46) with

$c=\frac{\omega}{1-q},\qquad a_{n}=\frac{1+(-1)^{n+1}}{(1-q)(a+b)}\,b,\qquad
b_{n}=[n]_{q}-a_{n},\qquad c_{n}=-cb_{n}.$

Indeed, the proof is by induction on $n$. For $n=1$, LHS of (46) gives

$(x-c)D_{q,\omega}P_{1}(x)=x-c,$

while RHS gives

$a_{1}P_{1}(x)+b_{1}(x-c)P_{0}(x)=(a_{1}+b_{1})(x-c)=x-c.$

Assuming (46) to hold, with $k$ instead of $n$, for $k=1,2,\dots,n$, we will
prove it for $k=n+1$. Apply $(x-c)D_{q,\omega}$ to (6) to obtain

$\displaystyle(x-c)D_{q,\omega}\big{(}(x-c)P_{n}(x)\big{)}=(x-c)D_{q,\omega}\big{(}P_{n+1}(x)+C_{n}P_{n-1}(x)\big{)}.$

Using the identity (see [3, (2.9)])

$D_{q,\omega}(fg)=g(qx+\omega)D_{q,\omega}f+fD_{q,\omega}g\qquad(f,g\in\mathcal{P}),$

we get

$\displaystyle(x-c)D_{q,\omega}P_{n+1}(x)$
$\displaystyle=q(x-c)^{2}D_{q,\omega}P_{n}(x)+(x-c)P_{n}(x)-(x-c)C_{n}D_{q,\omega}P_{n-1}(x).$

Now using successively (46), with $k$ instead of $n$, for $k=n$ and $k=n-1$
and (6) it follows that

$\displaystyle(x-c)D_{q,\omega}P_{n+1}(x)$
$\displaystyle=q(x-c)(a_{n}P_{n}(x)+b_{n}(x-c)P_{n-1}(x))$
$\displaystyle\quad+(x-c)P_{n}(x)-(a_{n-1}P_{n-1}(x)+b_{n-1}(x-c)P_{n-2}(x))C_{n}$
$\displaystyle=qa_{n}(P_{n+1}(x)+C_{n}P_{n-1}(x))+qb_{n}(x-c)(P_{n}(x)+C_{n-1}P_{n-2}(x))$
$\displaystyle\quad+(x-c)P_{n}(x)-(a_{n-1}P_{n-1}(x)+b_{n-1}(x-c)P_{n-2}(x))C_{n}$
$\displaystyle=qa_{n}P_{n+1}(x)+(qa_{n}-a_{n-1})C_{n}P_{n-1}(x)+(1+qb_{n})(x-c)P_{n}(x)$
$\displaystyle\quad+(qb_{n}C_{n-1}-b_{n-1}C_{n})(x-c)P_{n-2}(x)$
$\displaystyle=a_{n-1}P_{n+1}(x)+(1-a_{n-1}+q(a_{n}+b_{n}))(x-c)P_{n}(x)$
$\displaystyle\quad+(qb_{n}C_{n-1}-b_{n-1}C_{n})(x-c)P_{n-2}(x).$

The reader should convince himself that the following relations hold:
$a_{n-1}=a_{n+1}$, $1-a_{n-1}+q(a_{n}+b_{n})=b_{n+1}$, and
$qb_{n}C_{n-1}-b_{n-1}C_{n}=0$. Thus (46) holds for $n+1$, and our claim
follows. Note also that

$\displaystyle\qquad b_{2}=q+1,\qquad C_{1}=(a-b-(a+b)q)q,\qquad
C_{2}=(a+b)(1-q^{2})q^{2}.$

Under the notation of Theorem 5.2, we get

$\displaystyle\lambda_{\pm}$ $\displaystyle=c\pm(a-b-(a+b)q)^{1/2}q^{1/2},$

and so

$\displaystyle q(\omega+qc-\lambda_{+})(\omega+qc-\lambda_{-})$
$\displaystyle\quad=(b-a+(a+b)q)q^{2}\not=(-b-a+(a+b)q)q^{2}=-\frac{C_{2}}{b_{2}},$

because $\omega+q\,c=c$. Thus, from Theorem 5.2, the result follows. ∎

If we replace, in Theorem 3.1, the Askey-Wilson operator by the Hahn operator,
then ${\bf S}_{q}$ becomes the identity, as in the case of the standard
derivative, and Theorem 3.1 $i)$ reduces to ${\bf D}_{q,\omega}(\rho{\bf
u})=\psi{\bf u}$. In this context, an analogue of Theorem 3.1 appears in
Smaili’s PhD thesis under the supervision of Maroni (see [20, Theorem 1.1]).

## Acknowledgements

The authors thank T. H. Koornwinder for several helpful comments. The authors
also thank P. Maroni for his comments, and for offering them an original
printed copy of [21] during his last visit to Portugal. This work was
supported by the Centre for Mathematics of the University of Coimbra-
UIDB/00324/2020, funded by the Portuguese Government through FCT/ MCTES. The
first author thanks CMUP, University of Porto, for their support and
hospitality. The second author also thanks CMUP for their support. CMUP is a
member of LASI, which is financed by national funds through FCT, under the
projects with reference UIDB/00144/2020 and UIDP/00144/2020.

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