Special values for continuous $q$-Jacobi polynomials

Special values for the continuous $q$-Jacobi polynomials

Howard S. Cohl ${}^{\dagger}\\!\\!\ $ and Roberto S. Costas-Santos
${}^{\S}\\!\\!\ $ H. S. Cohl and R. S. Costas-Santos

† Applied and Computational Mathematics Division, National Institute of
Standards and Technology, Gaithersburg, MD 20899-8910, USA
http://www.nist.gov/itl/math/msg/howard-s-cohl.cfm<EMAIL_ADDRESS>§
Departamento de Métodos Cuantitativos, Universidad Loyola Andalucía, E-41704,
Dos Hermanas, Seville, Spain http://www.rscosan.com<EMAIL_ADDRESS>Received
???, in final form ????; Published online ???? We study special values for the
continuous $q$-Jacobi polynomials and present applications of these special
values which arise from bilinear generating functions, and in particular the
Poisson kernel for these polynomials.

##  1 Preliminaries

We adopt the following set notations:
$\mathbb{N}_{0}:=\\{0\\}\cup{\mathbb{N}}=\\{0,1,2,...\\}$, and we use the sets
$\mathbb{Z}$, $\mathbb{R}$, $\mathbb{C}$ which represent the integers, real
numbers and complex numbers respectively,
${{\mathbb{C}}^{\ast}}:={\mathbb{C}}\setminus\\{0\\}$, and
${{\mathbb{C}}^{\dagger}}:={{\mathbb{C}}^{\ast}}\setminus\\{z\in{\mathbb{C}}:|z|=1\\}$.
We also adopt the following multiset notation and conventions. Let ${\bf
a}:=\\{a,b,c,d\\}$, $a,b,c,d\in\mathbb{{{\mathbb{C}}^{\ast}}}$. Throughout the
paper, we assume that the empty sum vanishes and the empty product is unity.

######  Definition 1.1.

We adopt the following conventions for succinctly writing elements of lists.
To indicate sequential positive and negative elements, we write

$\pm a:=\\{a,-a\\}.$

We also adopt the analogous notations

${\mathrm{e}}^{\pm
i\theta}:=\\{{\mathrm{e}}^{i\theta},{\mathrm{e}}^{-i\theta}\\},\quad
z^{\pm}:=\\{z,z^{-1}\\}.$

Within a list of items, we define

$a+\left\\{\begin{subarray}{c}{x_{1}}\\\\[-3.41418pt] {\vdots}\\\
{x_{n}}\end{subarray}\right\\}:=\\{a+x_{1},\ldots,a+x_{n}\\}.$

In the same vein, consider the numbers $f_{s}\in\mathbb{C}$ with
$s\in{\mathcal{S}}\subset{\mathbb{N}}$, with ${\mathcal{S}}$ finite. Then, the
notation $\\{f_{s}\\}$ represents the multiset of all complex numbers $f_{s}$
such that $s\in{\mathcal{S}}$. Furthermore, consider some $p\in{\mathcal{S}}$,
then the notation $\\{f_{s}\\}_{s\neq p}$ represents the sequence of all
complex numbers $f_{s}$ such that $s\in{\mathcal{S}}\\!\setminus\\!\\{p\\}$.
In addition, for the empty list, $n=0$, we take

$\\{a_{1},{...},a_{n}\\}:=\emptyset.$

In this paper we will be using standard notations for both finite and
$q$-shifted factorials: including multi-$q$-shifted factorial notation where a
comma delineated list represents products (see [6, Appendix I]). We also adopt
standard notations for both terminating and nonterminating basic
hypergeometric functions ${}_{r}\phi_{s}$ [4, (17.4.1)], and a nice
specialization which is often referred to as very-well-poised basic
hypergeometric functions ${}_{r+1}W_{r}$ [6, (2.1.11)]. For those not familiar
with these functions, we urge the reader to refer to some important standard
references on the subject, and in particular [4, Chapter 17] and links therein
and the entire book by Gasper & Rahman (2004) [6]. For particulars relating to
orthogonal polynomials we suggest the reader to refer to [8, Chapters 1,9,14],
the monograph [7] and the Memoirs of AMS article by Askey & Wilson [2]. For a
general treatment of special functions, one should refer to [1] and [4].

###  1.1 The Askey–Wilson polynomials

The Askey–Wilson polynomials can be defined in terms of the terminating
balanced basic hypergeometric series [8, (14.1.1)]

$p_{n}(x;{\bf
a}|q):=a^{-n}(ab,ac,ad;q)_{n}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{n-1}abcd,az^{\pm}}{ab,ac,ad};q,q\right),$
(1)

where $x=\tfrac{1}{2}(z+z^{-1})$. In some of the derivations given below, we
use the renormalized version of the Askey–Wilson polynomials given by

$r_{n}(x;{\bf
a}|q):=\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{n-1}abcd,az^{\pm}}{ab,ac,ad};q,q\right)=\frac{a^{n}}{(ab,ac,ad;q)_{n}}\,p_{n}(x;{\bf
a}|q).$ (2)

The Askey–Wilson polynomials have the following special values [9, (114)]

$p_{n}(\tfrac{1}{2}(a+a^{-1});a,b,c,d|q)=a^{-n}(ab,ac,ad;q)_{n}$ (3)

(and similarly for arguments $\tfrac{1}{2}(b+b^{-1})$,
$\tfrac{1}{2}(c+c^{-1})$, $\tfrac{1}{2}(d+d^{-1})$).

###  1.2 The $q$-Racah polynomials

Let $m\in\mathbb{C}$, $n,N\in{\mathbb{N}}_{0}$ such that
$n\in\\{0,\ldots,N\\}$. Let us consider the $q$-Racah polynomials
$R_{n}(\mu(m);\bar{\alpha},\bar{\beta},\bar{\gamma},\bar{\delta}|q)$ which are
discrete cases of the Askey–Wilson polynomials (1) (note that we have used the
notation $\\{\bar{\alpha},\bar{\beta},\bar{\gamma},\bar{\delta}\\}$ instead of
the standard $\\{\alpha,\beta,\gamma,\delta\\}$ in order to disambiguate
between the $\\{\alpha,\beta\\}$ parameters which appear in the study of
continuous $q$-Jacobi polynomials (see §1.3 below)). The $q$-Racah polynomials
are defined by [8, (14.2.1)]

$R_{n}(\mu(m);\bar{\alpha},\bar{\beta},\bar{\gamma},\bar{\delta}|q):=\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{n+1}\bar{\alpha}\bar{\beta},q^{-m},q^{m+1}\bar{\gamma}\bar{\delta}}{q\bar{\alpha},q\bar{\beta}\bar{\delta},q\bar{\gamma}};q,q\right),$
(4)

where
$\mu(m):=\mu(m;{\bar{\gamma}},{\bar{\delta}}|q):=q^{-m}+q^{m+1}\bar{\gamma}\bar{\delta}$,
and

${\bar{\alpha}}=q^{-N-1}\quad\mbox{or}\quad{\bar{\beta}}{\bar{\delta}}=q^{-N-1}\quad\mbox{or}\quad{\bar{\gamma}}=q^{-N-1}.$
(5)

The $q$-Racah polynomials are orthogonal on the finite $q$-quadratic set
$\\{\mu(m;{\bar{\gamma}},{\bar{\delta}}|q)\\}$. See [8, (14.2.2)] for the
orthogonality relation for $q$-Racah polynomials. Observe that for
$m\in{\mathbb{N}}_{0}$, the $q$-Racah polynomials satisfy the following
duality relation [9, (146)]

$R_{n}(\mu(m);{\bar{\alpha}},{\bar{\beta}},{\bar{\gamma}},{\bar{\delta}}|q)=R_{m}(\mu(n);{\bar{\gamma}},{\bar{\delta}},{\bar{\alpha}},{\bar{\beta}}|q).$
(6)

###  1.3 The continuous q-Jacobi polynomials

The continuous $q$-Jacobi polynomials [8, Section 14.10] follow from the
Askey–Wilson polynomials as follows [8, (14.1.19) and p. 467]

$P_{n}^{(\alpha,\beta)}(x|q):=\frac{q^{(\frac{\alpha}{2}+\frac{1}{4})n}}{(q,-q^{\frac{\alpha+\beta}{2}+\frac{1}{2}\left\\{\begin{subarray}{c}{1}\\\
{2}\end{subarray}\right\\}};q)_{n}}p_{n}(x;{\bf a}|q),$ (7)

where

${\bf
a}:=\\{a,b,c,d\\}:=\left\\{q^{\frac{1}{2}\alpha+\frac{1}{4}\left\\{\begin{subarray}{c}{1}\\\
{3}\end{subarray}\right\\}},-q^{\frac{1}{2}\beta+\frac{1}{4}\left\\{\begin{subarray}{c}{1}\\\
{3}\end{subarray}\right\\}}\right\\}.$ (8)

For the coefficients $\\{a,b\\}=\\{\pm q^{\frac{1}{2}}\\}$,
$c={q^{\alpha+\frac{1}{2}}}$, $d=-{q^{\beta+\frac{1}{2}}}$, the renormalized
Askey–Wilson polynomials are related to the continuous $q$-Jacobi polynomials
with base $q^{2}$, namely

$r_{n}(x;\pm
q^{\frac{1}{2}},{q^{\alpha+\frac{1}{2}}},-{q^{\beta+\frac{1}{2}}}|q)=q^{-n\alpha}\frac{(q,-q^{\alpha+\beta+1};q)_{n}}{(q^{\alpha+1},-q^{\beta+1};q)_{n}}P_{n}^{(\alpha,\beta)}(x|{q^{2}}).$
(9)

In fact, this Askey–Wilson polynomial is related by a quadratic transformation
which follows from a formula due to Singh [2, (4.22)], [6, (III.21)], [11,
(7)]

$\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-2n},q^{2n}a^{2},c^{2},qb^{2}}{-a,-qa,q^{2}b^{2}c^{2}};q^{2},q^{2}\right)=(bc)^{n}\frac{(-q,-\frac{a}{bc};q)_{n}}{(-a,-qbc;q)_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{n}{a},\frac{c}{b},\frac{qb}{c}}{-q,-\frac{a}{bc},qbc};q,q\right),$
(10)

or equivalently,

$\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{n}a^{2},c^{2},q^{\frac{1}{2}}b^{2}}{-a,-q^{\frac{1}{2}}a,qb^{2}c^{2}};q,q\right)=(bc)^{n}\frac{(-q^{\frac{1}{2}},-\frac{a}{bc};q^{\frac{1}{2}})_{n}}{(-a,-q^{\frac{1}{2}}bc;q^{\frac{1}{2}})_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-\frac{n}{2}},q^{\frac{n}{2}}{a},\frac{c}{b},\frac{q^{\frac{1}{2}}b}{c}}{-q^{\frac{1}{2}},-\frac{a}{bc},q^{\frac{1}{2}}bc};q^{\frac{1}{2}},q^{\frac{1}{2}}\right),$
(11)

but was also proved independently by Askey and Wilson in [2, (3.2)]), [10,
(1.12)]

$r_{n}\left(x;\pm
q^{\frac{1}{2}},{q^{\alpha+\frac{1}{2}}},-{q^{\beta+\frac{1}{2}}}|q\right)=q^{-n\alpha}\frac{(-q^{\alpha+1},-q^{\alpha+\beta+1};q)_{n}}{(-q^{\beta+1},-q;q)_{n}}r_{n}\left(x;q^{\alpha+\frac{1}{2}\left\\{\begin{subarray}{c}{1}\\\
{3}\end{subarray}\right\\}},-q^{\beta+\frac{1}{2}\left\\{\begin{subarray}{c}{1}\\\
{3}\end{subarray}\right\\}}|q^{2}\right).$ (12)

The parity relation for continuous $q$-Jacobi polynomials is given by [9,
(165)]

$P_{n}^{(\alpha,\beta)}(-x|q)=-q^{\frac{1}{2}n(\alpha-\beta)}P_{n}^{(\beta,\alpha)}(x|q).$
(13)

The continuous $q$-Jacobi polynomials are orthogonal over
$x=\frac{1}{2}({\mathrm{e}}^{i\theta}+{\mathrm{e}}^{-i\theta})=\cos\theta\in(-1,1)$
which is demonstrated by the orthogonality relation

$\int_{-1}^{1}P_{m}^{(\alpha,\beta)}(x|q)P_{n}^{(\alpha,\beta)}(x|q)\,\frac{w(x;\alpha,\beta|q)}{\sqrt{1-x^{2}}}\,{\mathrm{d}}x=h_{n}(\alpha,\beta|q)\delta_{m,n},$
(14)

where the weight function and the norm for the continuous $q$-Jacobi
polynomials are given by

$w(x;\alpha,\beta|q)=\frac{({\mathrm{e}}^{\pm
2i\theta};q)_{\infty}}{(q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}}^{\pm
i\theta},q^{\frac{1}{2}\alpha+\frac{3}{4}}{\mathrm{e}}^{\pm
i\theta},-q^{\frac{1}{2}\beta+\frac{1}{4}}{\mathrm{e}}^{\pm
i\theta},-q^{\frac{1}{2}\beta+\frac{3}{4}}{\mathrm{e}}^{\pm
i\theta};q)_{\infty}},$ (15) $h_{n}(\alpha,\beta|q)=\frac{2\pi
q^{(\alpha+\frac{1}{2})n}(q^{\frac{\alpha+\beta+2}{2}},q^{\frac{\alpha+\beta+3}{2}};q)_{\infty}(q^{\alpha+1},q^{\beta+1},q^{\frac{\alpha+\beta+1}{2}};q)_{n}}{(q,q^{\alpha+1},q^{\beta+1},-q^{\frac{\alpha+\beta+1}{2}},-q^{\frac{\alpha+\beta+2}{2}};q)_{\infty}(q,q^{\alpha+\beta+1},q^{\frac{\alpha+\beta+3}{2}};q)_{n}}.$
(16)

######  Theorem 1.2.

Let $n\in{\mathbb{N}}_{0}$, $|q|<1$, $\alpha,\beta\in{\mathbb{C}}$,
$x=\frac{1}{2}(z+z^{-1})\in{\mathbb{C}}$. Then the exhaustive list of all
balanced ${}_{4}\phi_{3}$ continuous $q$-Jacobi polynomial representations are
given by

$\displaystyle
P_{n}^{(\alpha,\beta)}(x|q)=\frac{(q^{\alpha+1};q)_{n}}{(q;q)_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{\alpha+\beta+n+1},q^{\frac{1}{2}\alpha+\frac{1}{4}}z^{\pm}}{q^{\alpha+1},-q^{\frac{\alpha+\beta}{2}+\frac{1}{2}\left\\{\begin{subarray}{c}{1}\\\
{2}\end{subarray}\right\\}}};q,q\right)$ (17) $\displaystyle\hskip
14.22636pt=q^{-\frac{n}{2}}\frac{(q^{\alpha+1},-q^{\frac{\alpha+\beta+3}{2}};q)_{n}}{(q,-q^{\frac{\alpha+\beta+1}{2}};q)_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{\alpha+\beta+n+1},q^{\frac{1}{2}\alpha+\frac{3}{4}}z^{\pm}}{q^{\alpha+1},-q^{\frac{\alpha+\beta}{2}+\frac{1}{2}\left\\{\begin{subarray}{c}{2}\\\
{3}\end{subarray}\right\\}}};q,q\right)$ (18) $\displaystyle\hskip
14.22636pt=\left(-q^{\frac{\alpha-\beta}{2}}\right)^{n}\frac{(q^{\beta+1};q)_{n}}{(q;q)_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{\alpha+\beta+n+1},-q^{\frac{1}{2}\beta+\frac{1}{4}}z^{\pm}}{q^{\beta+1},-q^{\frac{\alpha+\beta}{2}+\frac{1}{2}\left\\{\begin{subarray}{c}{1}\\\
{2}\end{subarray}\right\\}}};q,q\right)$ (19) $\displaystyle\hskip
14.22636pt=\left(-q^{\frac{\alpha-\beta-1}{2}}\right)^{n}\frac{(q^{\beta+1},-q^{\frac{\alpha+\beta+3}{2}};q)_{n}}{(q,-q^{\frac{\alpha+\beta+1}{2}};q)_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{\alpha+\beta+n+1},-q^{\frac{1}{2}\beta+\frac{3}{4}}z^{\pm}}{q^{\beta+1},-q^{\frac{\alpha+\beta}{2}+\frac{1}{2}\left\\{\begin{subarray}{c}{2}\\\
{3}\end{subarray}\right\\}}};q,q\right);$ (20) $\displaystyle
P_{n}^{(\alpha,\beta)}(x|q)$
$\displaystyle=q^{-\binom{n}{2}}\left(-q^{-\frac{1}{2}}\right)^{n}\frac{\left(q^{\frac{\alpha+\beta}{2}+\frac{1}{2}\left\\{\begin{subarray}{c}{1}\\\
{2}\end{subarray}\right\\}},q^{\frac{1}{2}\alpha+\frac{3}{4}}z^{\pm};q\right)_{n}}{\left(q,q^{\alpha+\beta+1};q\right)_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{-\alpha-n},-q^{-\frac{\alpha+\beta}{2}-n-\frac{1}{2}\left\\{\begin{subarray}{c}{0}\\\
{1}\end{subarray}\right\\}}}{q^{-\alpha-\beta-2n},q^{-\frac{1}{2}\alpha+\frac{1}{4}-n}z^{\pm}};q,q\right)$
(21)
$\displaystyle=q^{-\binom{n}{2}}(-1)^{n}\frac{\left(q^{\frac{\alpha+\beta}{2}+\frac{1}{2}\left\\{\begin{subarray}{c}{1}\\\
{2}\end{subarray}\right\\}},q^{\frac{1}{2}\alpha+\frac{1}{4}}z^{\pm};q\right)_{n}}{\left(q,q^{\alpha+\beta+1};q\right)_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{-\alpha-n},-q^{-\frac{\alpha+\beta}{2}-n+\frac{1}{2}\left\\{\begin{subarray}{c}{0}\\\
{1}\end{subarray}\right\\}}}{q^{-\alpha-\beta-2n},q^{-\frac{1}{2}\alpha+\frac{3}{4}-n}z^{\pm}};q,q\right)$
(22)
$\displaystyle=\left(q^{\frac{\alpha-\beta-n}{2}}\right)^{n}\frac{\left(q^{\frac{\alpha+\beta}{2}+\frac{1}{2}\left\\{\begin{subarray}{c}{1}\\\
{2}\end{subarray}\right\\}},-q^{\frac{1}{2}\beta+\frac{3}{4}}z^{\pm};q\right)_{n}}{\left(q,q^{\alpha+\beta+1};q\right)_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{-\beta-n},-q^{-\frac{\alpha+\beta}{2}-n-\frac{1}{2}\left\\{\begin{subarray}{c}{0}\\\
{1}\end{subarray}\right\\}}}{q^{-\alpha-\beta-2n},-q^{-\frac{1}{2}\beta+\frac{1}{4}-n}z^{\pm}};q,q\right)$
(23)
$\displaystyle=q^{-\binom{n}{2}}\left(q^{\frac{\alpha-\beta}{2}}\right)^{n}\frac{\left(q^{\frac{\alpha+\beta}{2}+\frac{1}{2}\left\\{\begin{subarray}{c}{1}\\\
{2}\end{subarray}\right\\}},-q^{\frac{1}{2}\beta+\frac{1}{4}}z^{\pm};q\right)_{n}}{\left(q,q^{\alpha+\beta+1};q\right)_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{-\beta-n},-q^{-\frac{\alpha+\beta}{2}-n+\frac{1}{2}\left\\{\begin{subarray}{c}{0}\\\
{1}\end{subarray}\right\\}}}{q^{-\alpha-\beta-2n},-q^{-\frac{1}{2}\beta+\frac{3}{4}-n}z^{\pm}};q,q\right)\\!;$
(24)

and

$\displaystyle
P_{n}^{(\alpha,\beta)}(x|q)=z^{n}q^{(\frac{\alpha}{2}+\frac{1}{4})n}\frac{\left(q^{\alpha+1},-q^{\frac{1}{2}\beta+\frac{1}{4}\left\\{\begin{subarray}{c}{1}\\\
{3}\end{subarray}\right\\}}z^{-1};q\right)_{n}}{(q,-q^{\frac{\alpha+\beta}{2}+\frac{1}{2}\left\\{\begin{subarray}{c}{1}\\\
{2}\end{subarray}\right\\}};q)_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{-\beta-n},q^{\frac{1}{2}\alpha+\frac{1}{4}\left\\{\begin{subarray}{c}{1}\\\
{3}\end{subarray}\right\\}}z}{q^{\alpha+1},-q^{-\frac{1}{2}\beta-n+\frac{1}{4}\left\\{\begin{subarray}{c}{1}\\\
{3}\end{subarray}\right\\}}z};q,q\right)$ (25) $\displaystyle\hskip
2.84544pt=z^{n}q^{(\frac{\alpha}{2}+\frac{1}{4})n}\frac{\left(q^{\beta+1},q^{\frac{1}{2}\alpha+\frac{1}{4}\left\\{\begin{subarray}{c}{1}\\\
{3}\end{subarray}\right\\}}z^{-1};q\right)_{n}}{(q,-q^{\frac{\alpha+\beta}{2}+\frac{1}{2}\left\\{\begin{subarray}{c}{1}\\\
{2}\end{subarray}\right\\}};q)_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{-\alpha-n},-q^{\frac{1}{2}\beta+\frac{1}{4}\left\\{\begin{subarray}{c}{1}\\\
{3}\end{subarray}\right\\}}z}{q^{\beta+1},q^{-\frac{1}{2}\alpha-n+\frac{1}{4}\left\\{\begin{subarray}{c}{1}\\\
{3}\end{subarray}\right\\}}z};q,q\right)$ (26) $\displaystyle\hskip
2.84544pt=z^{n}q^{(\frac{\alpha}{2}+\frac{1}{4})n}\frac{\left(q^{\frac{\alpha}{2}+\frac{3}{4}}z^{-1},-q^{\frac{\beta}{2}+\frac{3}{4}}z^{-1};q\right)_{n}}{(q,-q^{\frac{\alpha+\beta+2}{2}};q)_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},-q^{-\frac{\alpha+\beta+1}{2}-n},q^{\frac{1}{2}\alpha+\frac{1}{4}}z,-q^{\frac{1}{2}\beta+\frac{1}{4}}z}{-q^{\frac{\alpha+\beta+1}{2}},q^{-\frac{1}{2}\alpha+\frac{1}{4}-n}z,-q^{-\frac{1}{2}\beta+\frac{1}{4}-n}z};q,q\right)$
(27) $\displaystyle\hskip
2.84544pt=z^{n}q^{(\frac{\alpha}{2}+\frac{1}{4})n}\frac{\left(q^{\frac{\alpha}{2}+\frac{3}{4}}z^{-1},-q^{\frac{\beta}{2}+\frac{1}{4}}z^{-1};q\right)_{n}}{(q,-q^{\frac{\alpha+\beta+1}{2}};q)_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},-q^{-\frac{\alpha+\beta}{2}-n},q^{\frac{1}{2}\alpha+\frac{1}{4}}z,-q^{\frac{1}{2}\beta+\frac{3}{4}}z}{-q^{\frac{\alpha+\beta+2}{2}},q^{-\frac{1}{2}\alpha+\frac{1}{4}-n}z,-q^{-\frac{1}{2}\beta+\frac{3}{4}-n}z};q,q\right)$
(28) $\displaystyle\hskip
2.84544pt=z^{n}q^{(\frac{\alpha}{2}+\frac{1}{4})n}\frac{\left(q^{\frac{\alpha}{2}+\frac{1}{4}}z^{-1},-q^{\frac{\beta}{2}+\frac{3}{4}}z^{-1};q\right)_{n}}{(q,-q^{\frac{\alpha+\beta+1}{2}};q)_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},-q^{-\frac{\alpha+\beta}{2}-n},q^{\frac{1}{2}\alpha+\frac{3}{4}}z,-q^{\frac{1}{2}\beta+\frac{1}{4}}z}{-q^{\frac{\alpha+\beta+2}{2}},q^{-\frac{1}{2}\alpha+\frac{3}{4}-n}z,-q^{-\frac{1}{2}\beta+\frac{1}{4}-n}z};q,q\right)$
(29) $\displaystyle\hskip
2.84544pt=z^{n}q^{(\frac{\alpha}{2}+\frac{1}{4})n}\frac{\left(-q^{\frac{\alpha+\beta+3}{2}},q^{\frac{\alpha}{2}+\frac{1}{4}}z^{-1},-q^{\frac{\beta}{2}+\frac{1}{4}}z^{-1};q\right)_{n}}{(q,-q^{\frac{\alpha+\beta+1}{2}},-q^{\frac{\alpha+\beta+2}{2}};q)_{n}}$
$\displaystyle\hskip
183.52019pt\times\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},-q^{-\frac{\alpha+\beta-1}{2}-n},q^{\frac{\alpha}{2}+\frac{3}{4}}z,-q^{\frac{\beta}{2}+\frac{3}{4}}z}{-q^{\frac{\alpha+\beta+3}{2}},q^{-\frac{\alpha}{2}+\frac{3}{4}-n}z,-q^{-\frac{\beta}{2}+\frac{3}{4}-n}z};q,q\right),$
(30)

and as well as the application of the map $z\mapsto z^{-1}$ in (25)–(30).

###### Proof.

In [3, Theorem 7, (13)–(15)] an exhaustive description of all balanced
${}_{4}\phi_{3}$ representations of the Askey–Wilson polynomials are given
with parameters ${\bf a}:=\\{a_{1},a_{2},a_{3},a_{4}\\}$. By using (7) which
is the description of the continuous $q$-Jacobi polynomials in terms of the
Askey–Wilson polynomials, one is able to obtain an exhaustive list of all
balanced ${}_{4}\phi_{3}$ representations of the continuous $q$-Jacobi
polynomials. In line with (7), we choose

${\bf
a}:=\left\\{q^{\frac{1}{2}\alpha+\frac{1}{4}},q^{\frac{1}{2}\alpha+\frac{3}{4}},-q^{\frac{1}{2}\beta+\frac{1}{4}},-q^{\frac{1}{2}\beta+\frac{3}{4}}\right\\}.$
(31)

By evaluating all permutations of (31) in [3, (13), (14), (15)], one may
obtain (17)–(20), (23)–(24), (25)–(30) respectively. Since the continuous
$q$-Jacobi polynomials are a function of $x=\frac{1}{2}(z+z^{-1})$, and
(17)–(24) are invariant under the interchange of $z\mapsto z^{-1}$, one may as
well apply this replacement to (25)–(30) to obtain six alternative
representations of the continuous $q$-Jacobi polynomials. This completes the
proof. ∎

######  Remark 1.3.

Note that one may also obtain an exhaustive list of all continuous $q$-Jacobi
polynomial terminating ${}_{8}W_{7}$ representations by applying the above
procedure to [3, (16)–(19)]. However, we will omit this computation for the
present work. We might also add that by counting the members of the
equivalence classes of terminating ${}_{4}\phi_{3}$ representations of the
continuous $q$-Jacobi polynomials, one may explore the symmetry group of these
transformations which should necessarily be a subgroup of the symmetric group
$S_{6}$, the symmetry group of the terminating representations of the
Askey–Wilson polynomials, see [3, Section 4], and references therein.

##  2 Specialization values of continuous $q$-Jacobi polynomials

For the following special values for the argument of the continuous $q$-Jacobi
polynomials, we are able to re-express them in terms of Askey–Wilson
polynomials with degree $m$ as opposed to $n$.

######  Theorem 2.1.

Let $n,m\in{\mathbb{N}}_{0}$, $q\in{{\mathbb{C}}^{\dagger}}$,
$\alpha,\beta\in\mathbb{C}$. The following continuous $q$-Jacobi polynomial
specializations have alternative Askey–Wilson representations given as follows

$\displaystyle
P_{n}^{(\alpha,\beta)}(\tfrac{1}{2}(q^{\frac{1}{2}\alpha+\frac{1}{4}+m}+q^{-\frac{1}{2}\alpha-\frac{1}{4}-m})|q)=\frac{(q^{\alpha+1};q)_{n}\left(q^{\frac{1}{2}(\alpha+\beta+1)}\right)^{m}}{(q;q)_{n}(q^{\alpha+1},-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)};q)_{m}}$
$\displaystyle\hskip 99.58464pt\times
p_{m}(\tfrac{1}{2}(q^{\frac{\alpha+\beta+1}{2}+n}+q^{-\frac{\alpha+\beta+1}{2}-n});q^{\frac{1}{2}(\alpha+\beta+1)},q^{\frac{1}{2}(\alpha-\beta+1)},-q^{\frac{1}{2}},-1|q),$
(32) $\displaystyle
P_{n}^{(\alpha,\beta)}(\tfrac{1}{2}(q^{\frac{1}{2}\alpha+\frac{3}{4}+m}+q^{-\frac{1}{2}\alpha-\frac{3}{4}-m})|q)=\frac{q^{-\frac{n}{2}}\left(q^{\frac{1}{2}(\alpha+\beta+1)}\right)^{m}(q^{\alpha+1},-q^{\frac{\alpha+\beta+3}{2}};q)_{n}}{(q,-q^{\frac{\alpha+\beta+1}{2}};q)_{n}(q^{\alpha+1},-q^{\frac{1}{2}(\alpha+\beta+2)},-q^{\frac{1}{2}(\alpha+\beta+3)};q)_{m}}$
$\displaystyle\hskip 99.58464pt\times
p_{m}(\tfrac{1}{2}(q^{\frac{\alpha+\beta+1}{2}+n}+q^{-\frac{\alpha+\beta+1}{2}-n});q^{\frac{1}{2}(\alpha+\beta+1)},q^{\frac{1}{2}(\alpha-\beta+1)},-q^{\frac{1}{2}},-q|q),$
(33) $\displaystyle
P_{n}^{(\alpha,\beta)}(-\tfrac{1}{2}(q^{\frac{1}{2}\beta+\frac{1}{4}+m}+q^{-\frac{1}{2}\beta-\frac{1}{4}-m})|q)=\frac{\left(-q^{\frac{1}{2}(\alpha-\beta)}\right)^{n}\left(q^{\frac{1}{2}(\alpha+\beta+1)}\right)^{m}(q^{\beta+1};q)_{n}}{(q;q)_{n}(q^{\beta+1},-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)};q)_{m}}$
$\displaystyle\hskip 99.58464pt\times
p_{m}(\tfrac{1}{2}(q^{\frac{\alpha+\beta+1}{2}+n}+q^{-\frac{\alpha+\beta+1}{2}-n});q^{\frac{1}{2}(\alpha+\beta+1)},q^{\frac{1}{2}(\beta-\alpha+1)},-q^{\frac{1}{2}},-1|q),$
(34) $\displaystyle
P_{n}^{(\alpha,\beta)}(-\tfrac{1}{2}(q^{\frac{1}{2}\beta+\frac{3}{4}+m}+q^{-\frac{1}{2}\beta-\frac{3}{4}-m})|q)=\frac{\left(-q^{\frac{1}{2}(\alpha-\beta-1)}\right)^{n}\left(q^{\frac{1}{2}(\alpha+\beta+1)}\right)^{m}(q^{\beta+1},-q^{\frac{1}{2}(\alpha+\beta+3)};q)_{n}}{(q,-q^{\frac{1}{2}(\alpha+\beta+1)};q)_{n}(q^{\beta+1},-q^{\frac{1}{2}(\alpha+\beta+2)},-q^{\frac{1}{2}(\alpha+\beta+3)};q)_{m}}$
$\displaystyle\hskip 99.58464pt\times
p_{m}(\tfrac{1}{2}(q^{\frac{\alpha+\beta+1}{2}+n}+q^{-\frac{\alpha+\beta+1}{2}-n});q^{\frac{1}{2}(\alpha+\beta+1)},q^{\frac{1}{2}(\beta-\alpha+1)},-q^{\frac{1}{2}},-q|q).$
(35)

###### Proof.

Start by considering the Askey–Wilson polynomial representations of these
continuous $q$-Jacobi polynomials (7) with these particular arguments. The
specific arguments provided along with the identification that
$x=\frac{1}{2}(z+z^{-1})$ and therefore

$z=z_{m}\in\\{q^{\frac{1}{2}\alpha+\frac{1}{4}+m},q^{\frac{1}{2}\alpha+\frac{3}{4}+m},-q^{\frac{1}{2}\beta+\frac{1}{4}+m},-q^{\frac{1}{2}\beta+\frac{3}{4}+m}\\},$
(36)

for (2.1)–(2.1) respectively. Providing the particular specializations of the
argument produces the following ${}_{4}\phi_{3}$ representations

$\displaystyle
P_{n}^{(\alpha,\beta)}(\tfrac{1}{2}(q^{\frac{\alpha}{2}+\frac{1}{4}+m}+q^{-\frac{\alpha}{2}-\frac{1}{4}-m})|q)=\frac{(q^{\alpha+1};q)_{n}}{(q;q)_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{\alpha+\beta+1+n},q^{-m},q^{\alpha+\frac{1}{2}+m}}{q^{\alpha+1},-q^{\frac{\alpha+\beta+1}{2}},-q^{\frac{\alpha+\beta+2}{2}}};q,q\right),$
(37) $\displaystyle
P_{n}^{(\alpha,\beta)}(\tfrac{1}{2}(q^{\frac{\alpha}{2}+\frac{3}{4}+m}+q^{-\frac{\alpha}{2}-\frac{3}{4}-m})|q)$
$\displaystyle\hskip
8.5359pt=q^{-\frac{n}{2}}\frac{(q^{\alpha+1},-q^{\frac{\alpha+\beta+3}{2}};q)_{n}}{(q,-q^{\frac{\alpha+\beta+1}{2}};q)_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{\alpha+\beta+1+n},q^{-m},q^{\alpha+\frac{3}{2}+m}}{q^{\alpha+1},-q^{\frac{\alpha+\beta+2}{2}},-q^{\frac{\alpha+\beta+3}{2}}};q,q\right),$
(38) $\displaystyle
P_{n}^{(\alpha,\beta)}(-\tfrac{1}{2}(q^{\frac{\beta}{2}+\frac{1}{4}+m}+q^{-\frac{\beta}{2}-\frac{1}{4}-m})|q)$
$\displaystyle\hskip
8.5359pt=\left(-q^{\frac{\alpha-\beta}{2}}\right)^{n}\frac{(q^{\beta+1};q)_{n}}{(q;q)_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{\alpha+\beta+1+n},q^{-m},q^{m+\beta+\frac{1}{2}}}{q^{\beta+1},-q^{\frac{\alpha+\beta+1}{2}},-q^{\frac{\alpha+\beta+2}{2}}};q,q\right),$
(39) $\displaystyle
P_{n}^{(\alpha,\beta)}(-\tfrac{1}{2}(q^{\frac{\beta}{2}+\frac{3}{4}+m}+q^{-\frac{\beta}{2}-\frac{3}{4}-m})|q)$
$\displaystyle\hskip
8.5359pt=\left(-q^{\frac{\alpha-\beta-1}{2}}\right)^{n}\frac{(q^{\beta+1},-q^{\frac{\alpha+\beta+3}{2}};q)_{n}}{(q,-q^{\frac{\alpha+\beta+1}{2}};q)_{n}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-n},q^{\alpha+\beta+1+n},q^{-m},q^{\beta+\frac{3}{2}+m}}{q^{\beta+1},-q^{\frac{\alpha+\beta+2}{2}},-q^{\frac{\alpha+\beta+3}{2}}};q,q\right).$
(40)

It is then straightforward to convert these particular continuous $q$-Jacobi
polynomials into a form where they can be represented as either Askey–Wilson
polynomials in the degree $n$ but also as Askey–Wilson polynomials in degree
$m$. Solving for the particular values of the constants ${\bf a}$ in (1) for
the degree $m$ case completes the proof. ∎

One, therefore, has the following special values for the continuous $q$-Jacobi
polynomials which also follow directly from the identity (3).

######  Corollary 2.2.

Let $n,m\in{\mathbb{N}}_{0}$, $q\in{{\mathbb{C}}^{\dagger}}$,
$\alpha,\beta\in\mathbb{C}$. Then

$\displaystyle
P_{n}^{(\alpha,\beta)}(\tfrac{1}{2}(q^{\frac{1}{2}\alpha+\frac{1}{4}}+q^{-\frac{1}{2}\alpha-\frac{1}{4}})|q)=\frac{(q^{\alpha+1};q)_{n}}{(q;q)_{n}},$
(41) $\displaystyle
P_{n}^{(\alpha,\beta)}(\tfrac{1}{2}(q^{\frac{1}{2}\alpha+\frac{3}{4}}+q^{-\frac{1}{2}\alpha-\frac{3}{4}})|q)=q^{-\frac{n}{2}}\frac{(q^{\alpha+1},-q^{\frac{1}{2}(\alpha+\beta+3)};q)_{n}}{(q,-q^{\frac{1}{2}(\alpha+\beta+1)};q)_{n}},$
(42) $\displaystyle
P_{n}^{(\alpha,\beta)}(-\tfrac{1}{2}(q^{\frac{1}{2}\beta+\frac{1}{4}}+q^{-\frac{1}{2}\beta-\frac{1}{4}})|q)=\left(-q^{\frac{\alpha-\beta}{2}}\right)^{n}\frac{(q^{\beta+1};q)_{n}}{(q;q)_{n}},$
(43) $\displaystyle
P_{n}^{(\alpha,\beta)}(-\tfrac{1}{2}(q^{\frac{1}{2}\beta+\frac{3}{4}}+q^{-\frac{1}{2}\beta-\frac{3}{4}})|q)=\left(-q^{\frac{\alpha-\beta-1}{2}}\right)^{n}\frac{(q^{\beta+1},-q^{\frac{1}{2}(\alpha+\beta+3)};q)_{n}}{(q,-q^{\frac{1}{2}(\alpha+\beta+1)};q)_{n}}.$
(44)

###### Proof.

Taking $m=0$ values in Theorem 2.1 completes the proof. ∎

######  Remark 2.3.

One may inquire regarding the computation of perhaps product formulas for

$\displaystyle
P_{n}^{(\alpha,\beta)}(-\tfrac{1}{2}(q^{\frac{1}{2}\alpha+\frac{1}{4}}+q^{-\frac{1}{2}\alpha-\frac{1}{4}})|q),P_{n}^{(\alpha,\beta)}(-\tfrac{1}{2}(q^{\frac{1}{2}\alpha+\frac{3}{4}}+q^{-\frac{1}{2}\alpha-\frac{3}{4}})|q),$
$\displaystyle
P_{n}^{(\alpha,\beta)}(\tfrac{1}{2}(q^{\frac{1}{2}\beta+\frac{1}{4}}+q^{-\frac{1}{2}\beta-\frac{1}{4}})|q),P_{n}^{(\alpha,\beta)}(\tfrac{1}{2}(q^{\frac{1}{2}\beta+\frac{3}{4}}+q^{-\frac{1}{2}\beta-\frac{3}{4}})|q),$

using symmetry (13) and the special values for the Askey–Wilson polynomials
given [9, (114)]. However, this is not possible since the interchange of
$\alpha\leftrightarrow\beta$ with the invariance of the argument prevents the
${}_{4}\phi_{3}$s from being summable.

One may also express the results of Theorem 2.1 in terms of $q$-Racah
polynomials (4).

######  Theorem 2.4.

Let $q\in{{\mathbb{C}}^{\dagger}}$. Then the specialization for the continuous
$q$-Jacobi polynomials there are the following special values given in terms
of $q$-Racah polynomials as follows

$\displaystyle
P_{n}^{(\alpha,\beta)}(\tfrac{1}{2}(q^{\frac{\alpha}{2}+\frac{1}{4}+m}+q^{-\frac{\alpha}{2}-\frac{1}{4}m})|q)$
$\displaystyle\hskip
8.5359pt=\frac{(q^{\alpha+1};q)_{n}}{(q;q)_{n}}R_{n}(q^{\alpha+\frac{1}{2}+m}+q^{-m};q^{\alpha},q^{\beta},-q^{\frac{\alpha+\beta}{2}},-q^{\frac{\alpha-\beta-1}{2}}|q)$
(45) $\displaystyle\hskip
8.5359pt=\frac{(q^{\alpha+1};q)_{n}}{(q;q)_{n}}R_{m}(q^{\alpha+\beta+1+n}+q^{-n};-q^{\frac{\alpha+\beta}{2}},-q^{\frac{\alpha-\beta-1}{2}},q^{\alpha},q^{\beta}|q),$
(46) $\displaystyle
P_{n}^{(\alpha,\beta)}(\tfrac{1}{2}(q^{\frac{\alpha}{2}+\frac{3}{4}+m}+q^{-\frac{\alpha}{2}-\frac{3}{4}-m})|q)$
$\displaystyle\hskip
8.5359pt=q^{-\frac{n}{2}}\frac{(q^{\alpha+1},-q^{\frac{\alpha+\beta+3}{2}};q)_{n}}{(q,-q^{\frac{\alpha+\beta+1}{2}};q)_{n}}R_{n}(q^{\alpha+\frac{3}{2}+m}+q^{-m};q^{\alpha},q^{\beta},-q^{\frac{\alpha+\beta+1}{2}},-q^{\frac{\alpha-\beta}{2}}|q)$
(47) $\displaystyle\hskip
8.5359pt=q^{-\frac{n}{2}}\frac{(q^{\alpha+1},-q^{\frac{\alpha+\beta+3}{2}};q)_{n}}{(q,-q^{\frac{\alpha+\beta+1}{2}};q)_{n}}R_{m}(q^{\alpha+\beta+1+n}+q^{-n};-q^{\frac{\alpha+\beta+1}{2}},-q^{\frac{\alpha-\beta}{2}},q^{\alpha},q^{\beta}|q),$
(48) $\displaystyle
P_{n}^{(\alpha,\beta)}(-\tfrac{1}{2}(q^{\frac{\beta}{2}+\frac{1}{4}+m}+q^{-\frac{\beta}{2}-\frac{1}{4}-m})|q)$
$\displaystyle\hskip
8.5359pt=\left(-q^{\frac{\alpha-\beta}{2}}\right)^{n}\frac{(q^{\beta+1};q)_{n}}{(q;q)_{n}}R_{n}(q^{\beta+\frac{1}{2}+m}+q^{-m};q^{\beta},q^{\alpha},-q^{\frac{\beta-\alpha-1}{2}},-q^{\frac{\alpha+\beta}{2}}|q)$
(49) $\displaystyle\hskip
8.5359pt=\left(-q^{\frac{\alpha-\beta}{2}}\right)^{n}\frac{(q^{\beta+1};q)_{n}}{(q;q)_{n}}R_{m}(q^{\alpha+\beta+1+n}+q^{-n};-q^{\frac{\alpha+\beta}{2}},-q^{\frac{\beta-\alpha-1}{2}},q^{\beta},q^{\alpha}|q),$
(50) $\displaystyle
P_{n}^{(\alpha,\beta)}(-\tfrac{1}{2}(q^{\frac{\beta}{2}+\frac{3}{4}+m}+q^{-\frac{\beta}{2}-\frac{3}{4}-m})|q)$
$\displaystyle\hskip
8.5359pt=\left(-q^{\frac{\alpha-\beta-1}{2}}\right)^{n}\frac{(q^{\beta+1},-q^{\frac{\alpha+\beta+3}{2}};q)_{n}}{(q,-q^{\frac{\alpha+\beta+1}{2}};q)_{n}}R_{n}(q^{\beta+\frac{3}{2}+m}+q^{-m};q^{\beta},q^{\alpha},-q^{\frac{\beta+\alpha+1}{2}},-q^{\frac{\beta-\alpha}{2}}|q)$
(51) $\displaystyle\hskip
8.5359pt=\left(-q^{\frac{\alpha-\beta-1}{2}}\right)^{n}\frac{(q^{\beta+1},-q^{\frac{\alpha+\beta+3}{2}};q)_{n}}{(q,-q^{\frac{\alpha+\beta+1}{2}};q)_{n}}R_{m}(q^{\alpha+\beta+1+n}+q^{-n};-q^{\frac{\alpha+\beta+1}{2}},-q^{\frac{\beta-\beta}{2}},q^{\beta},q^{\alpha}|q).$
(52)

###### Proof.

Start with Theorem 2.1 and utilize (37)–(40) with (4) to write these
hypergeometric representations in terms of $q$-Racah polynomials. This
completes the proof. ∎

######  Corollary 2.5.

Let $m\in{\mathbb{N}}_{0}$. If $\beta=-\frac{1}{2}$, $\beta=\frac{1}{2}$,
$\alpha=-\frac{1}{2}$, $\alpha=\frac{1}{2}$, respectively for the following
specialized continuous $q$-Jacobi polynomials

$\displaystyle
P_{n}^{(\alpha,\beta)}(\tfrac{1}{2}(q^{\frac{1}{2}\alpha+\frac{1}{4}+m}+q^{-\frac{1}{2}\alpha-\frac{1}{4}-m})|q),P_{n}^{(\alpha,\beta)}(\tfrac{1}{2}(q^{\frac{1}{2}\alpha+\frac{3}{4}+m}+q^{-\frac{1}{2}\alpha-\frac{3}{4}-m})|q),$
$\displaystyle
P_{n}^{(\alpha,\beta)}(-\tfrac{1}{2}(q^{\frac{1}{2}\beta+\frac{1}{4}+m}+q^{-\frac{1}{2}\beta-\frac{1}{4}-m})|q),P_{n}^{(\alpha,\beta)}(-\tfrac{1}{2}(q^{\frac{1}{2}\beta+\frac{3}{4}+m}+q^{-\frac{1}{2}\beta-\frac{3}{4}-m})|q),$

then one has the following duality relations

$\displaystyle
R_{n}(q^{\alpha+\frac{1}{2}+m}+q^{-m};q^{\alpha},q^{-\frac{1}{2}},-q^{\frac{1}{2}\alpha\pm\frac{1}{4}}|q)=R_{m}(q^{\alpha+\frac{1}{2}+n}+q^{-n};q^{\alpha},q^{-\frac{1}{2}},-q^{\frac{1}{2}\alpha\pm\frac{1}{4}}|q).$
(53)

###### Proof.

Replace $\beta=\pm\tfrac{1}{2}$, $\alpha=\pm\tfrac{1}{2}$ in Theorem 2.4
completes the proof. ∎

Now consider the continuous $q$-Jacobi polynomials with special argument

$x_{m}^{\pm}:=\pm\tfrac{1}{2}(q^{\frac{1}{4}+\frac{m}{2}}\\!+\\!q^{-\frac{1}{4}-\frac{m}{2}}),$
(54)

which is given in terms the Askey–Wilson polynomials through (7) as

$\displaystyle
P_{n}^{(\alpha,\beta)}(x_{m}^{\pm}|q)=\frac{\left(q^{\frac{1}{2}\alpha+\frac{1}{4}}\right)^{n}}{(q,-q^{\frac{\alpha+\beta+1}{2}},-q^{\frac{\alpha+\beta+2}{2}};q)_{n}}p_{n}(x_{m}^{\pm};q^{\frac{\alpha}{2}+\frac{1}{4}\left\\{\begin{subarray}{c}{1}\\\
{3}\end{subarray}\right\\}},-q^{\frac{\beta}{2}+\frac{1}{4}\left\\{\begin{subarray}{c}{1}\\\
{3}\end{subarray}\right\\}}|q).$ (55)

We consider some of the properties of the continuous $q$-Jacobi polynomials
with this argument. First, we show some alternative Askey–Wilson
representations of these polynomials with the special argument $x_{m}^{\pm}$.

######  Theorem 2.6.

Let $m,n\in{\mathbb{N}}_{0}$, $q\in{{\mathbb{C}}^{\dagger}}$,
$\alpha,\beta\in\mathbb{C}$. Then

$\displaystyle P_{n}^{(\alpha,\beta)}(x_{m}^{\pm}|q)=\frac{\left(\pm
q^{\frac{1}{2}\alpha+\frac{1}{4}}\right)^{n}}{(q^{\frac{1}{2}},-q^{\frac{1}{2}},-q^{\frac{\alpha+\beta+1}{2}};q^{\frac{1}{2}})_{n}}p_{n}(x_{m}^{\pm};q^{\frac{1}{4}},-q^{\frac{1}{4}},\pm
q^{\frac{\alpha}{2}+\frac{1}{4}},\mp
q^{\frac{\beta}{2}+\frac{1}{4}}|q^{\frac{1}{2}}),$ (56) $\displaystyle\hskip
19.91684pt=\frac{\left(\pm
q^{\frac{1}{2}\alpha}\right)^{n}\left(q^{\frac{\alpha+\beta+1}{4}}\right)^{m}(\pm
q^{\frac{\alpha+1}{2}},\mp
q^{\frac{\beta+1}{2}};q^{\frac{1}{2}})_{n}}{(q^{\frac{1}{2}},-q^{\frac{\alpha+\beta+1}{2}};q^{\frac{1}{2}})_{n}(-q^{\frac{1}{2}},\pm
q^{\frac{\alpha+1}{2}},\mp q^{\frac{\beta+1}{2}};q^{\frac{1}{2}})_{m}}$
$\displaystyle\hskip 48.36958pt\times
p_{m}(\tfrac{1}{2}(q^{\frac{\alpha+\beta+1+2n}{4}}\\!+\\!q^{-\frac{\alpha+\beta+1+2n}{4}});q^{\frac{\alpha+\beta+1}{4}},-q^{\frac{-\alpha-\beta+1}{4}},\pm
q^{\frac{\alpha-\beta+1}{4}},\mp
q^{\frac{\beta-\alpha+1}{4}}|q^{\frac{1}{2}}).$ (57)

###### Proof.

Start with the representation of the continuous $q$-Jacobi polynomials in
terms of the Askey–Wilson polynomials (7) with special argument $x_{m}^{\pm}$
(54), namely (55). Now consider $x=\frac{1}{2}(z+z^{-1})$, therefore
$z=z_{m}:=\pm q^{\frac{1}{4}+\frac{m}{2}}$ in (17) and replacing $q\mapsto
q^{2}$ one obtains,

$\displaystyle
P_{n}^{(\alpha,\beta)}(\pm\tfrac{1}{2}(q^{\frac{1}{2}+{m}}\\!+\\!q^{-\frac{1}{2}-{m}})|q^{2})=\frac{(q^{2\alpha+2};q^{2})_{n}}{(q^{2};q^{2})_{n}}$
$\displaystyle\hskip
156.49014pt\times\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q^{-2n},q^{2\alpha+2\beta+2+2n},\pm
q^{\alpha+1+m},\pm
q^{\alpha-m}}{q^{2\alpha+2},-q^{\alpha+\beta+1},-q^{\alpha+\beta+2}};q^{2},q^{2}\right).$
(58)

Applying the quadratic transformation (10) produces

$P_{n}^{(\alpha,\beta)}(\pm\tfrac{1}{2}(q^{\frac{1}{2}+{m}}+q^{-\frac{1}{2}-{m}})|q^{2})\\!=\\!(\pm
q^{\alpha})^{n}\frac{(\pm q^{\alpha+1},\mp
q^{\beta+1};q)_{n}}{(q,-q^{\alpha+\beta+1};q)_{n}}\\!\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{\\!q^{-n},q^{\alpha+\beta+1+n},q^{-m},q^{m+1}}{\pm
q^{\alpha+1},\mp q^{\beta+1},-q\\!};q,q\\!\right)\\!,$

which can be viewed as either an Askey–Wilson polynomial with degree $n$ or
$m$. Obtaining these representations through (1), and then replacing
$q^{2}\mapsto q$, completes the proof. ∎

######  Remark 2.7.

Note that the above proof could be accomplished directly using (11) in (17).
Furthermore one should observe that of all the ${}_{4}\phi_{3}$
representations (17)–(30), only (17) and (19) allow for the quadratic
transformation (10) (or (11)). However, if one starts with (19) in order to
prove Theorem 2.6, one arrives at an identical result. The fact that (17) and
(19) are the only representations which satisfy the quadratic transformation
(10) (or (11)) can be seen since (21)–(19) do not contain the necessary
$q^{n}$ numerator entry and (18), (20) do not satisfy the conditions given in
(11).

Now consider the $m=0,1$ special cases. This leads to the following result.

######  Corollary 2.8.

Let $n\in{\mathbb{N}}_{0}$, $q\in{{\mathbb{C}}^{\dagger}}$,
$\alpha,\beta\in\mathbb{C}$. Then

$\displaystyle
P_{n}^{(\alpha,\beta)}(\tfrac{1}{2}(q^{\frac{1}{2}}+q^{-\frac{1}{2}})|q^{2})$
$\displaystyle=$
$\displaystyle\left(q^{{\alpha}}\right)^{n}\frac{(q^{{\alpha+1}},-q^{{\beta+1}};q)_{n}}{(q,-q^{\alpha+\beta+1};q)_{n}},$
(59) $\displaystyle
P_{n}^{(\alpha,\beta)}(-\tfrac{1}{2}(q^{\frac{1}{2}}+q^{-\frac{1}{2}})|q^{2})$
$\displaystyle=$
$\displaystyle\left(-q^{\alpha}\right)^{n}\frac{(-q^{{\alpha+1}},q^{{\beta+1}};q)_{n}}{(q,-q^{{\alpha+\beta+1}};q)_{n}},$
(60) $\displaystyle
P_{n}^{(\alpha,\beta)}(\tfrac{1}{2}(q^{\alpha+\frac{1}{2}}+q^{-\alpha-\frac{1}{2}}|q^{2})$
$\displaystyle=$
$\displaystyle\frac{(q^{\alpha+1},-q^{\alpha+1};q)_{n}}{(q,-q;q)_{n}},$ (61)
$\displaystyle
P_{n}^{(\alpha,\beta)}(-\tfrac{1}{2}(q^{\beta+\frac{1}{2}}+q^{-\beta-\frac{1}{2}})|q^{2})$
$\displaystyle=$
$\displaystyle\left(-q^{\alpha-\beta}\right)^{n}\frac{(q^{\beta+1},-q^{\beta+1};q)_{n}}{(q,-q;q)_{n}}.$
(62)

###### Proof.

For (59), (60) use the sum [2, (4.23)] due to the quadratic transformation for
Askey–Wilson polynomials (10). ∎

######  Corollary 2.9.

Let $n\in{\mathbb{N}}_{0}$, $q\in{{\mathbb{C}}^{\dagger}}$,
$\alpha,\beta\in\mathbb{C}$. Then one also has the following special value

$\displaystyle
P_{n}^{(\alpha,\beta)}(\pm\tfrac{1}{2}(q^{\frac{3}{4}}+q^{-\frac{3}{4}})|q)=\left(\pm
q^{\frac{\alpha}{2}}\right)^{n}\frac{(\pm q^{\frac{\alpha+1}{2}},\mp
q^{\frac{\beta+1}{2}};q^{\frac{1}{2}})_{n}}{(q^{\frac{1}{2}},-q^{\frac{\alpha+\beta+1}{2}};q^{\frac{1}{2}})_{n}}$
$\displaystyle\hskip
147.95424pt\times\left(1-\frac{(1-q^{\frac{1}{2}})(1-q^{-\frac{n}{2}})(1-q^{\frac{\alpha+\beta+n+1}{2})}}{(1\mp
q^{\frac{\alpha+1}{2}})(1\pm q^{\frac{\beta+1}{2}})}\right).$ (63)

###### Proof.

The proof follows in exactly the same way as for the argument
$\pm\frac{1}{2}(q^{\frac{1}{4}}+q^{-\frac{1}{4}})$, except instead of using
the sum [2, (4.23)], directly use the quadratic transformation for
Askey–Wilson polynomials [2, (4.22)]. Then with this transformation, the
resulting terminating ${}_{4}\phi_{3}$ has a $q^{-1}$ as one of the numerator
parameters, so the sum truncates to the first two terms and the above
specializations follow. As well, simply using Theorem 2.6 with $m=1$ in (57)
completes the proof. ∎

######  Theorem 2.10.

Let $m,n,N\in{\mathbb{N}}_{0}$, $q\in{{\mathbb{C}}^{\dagger}}$,
$\alpha,\beta\in\mathbb{C}$, and in the following special values for the
continuous $q$-Jacobi polynomials for the positive sign in the argument,
choose $\alpha=-N-1$ and leave $\beta$ unrestricted and for the negative sign
in the argument chose $\beta=-N-1$ and $\alpha$ unrestricted. Furthermore, for
(64) let $n\leq N$, and for (65) let $m\leq N$. Then

$\displaystyle
P_{n}^{(\alpha,\beta)}(\pm\tfrac{1}{2}(q^{\frac{1}{4}+\frac{m}{2}}\\!+\\!q^{-\frac{1}{4}-\frac{m}{2}})|q)$
$\displaystyle=(\pm q^{\frac{1}{2}\alpha})^{n}\frac{(\pm
q^{\frac{\alpha+1}{2}},\mp
q^{\frac{\beta+1}{2}};q)_{n}}{(q^{\frac{1}{2}},-q^{\frac{\alpha+\beta+1}{2}};q)_{n}}R_{n}(q^{-\frac{m}{2}}+q^{\frac{m+1}{2}};\pm
q^{\frac{1}{2}\alpha},\pm q^{\frac{1}{2}\beta},-1,-1|q^{\frac{1}{2}})$ (64)
$\displaystyle=(\pm q^{\frac{1}{2}\alpha})^{n}\frac{(\pm
q^{\frac{\alpha+1}{2}},\mp
q^{\frac{\beta+1}{2}};q)_{n}}{(q^{\frac{1}{2}},-q^{\frac{\alpha+\beta+1}{2}};q)_{n}}R_{m}(q^{-\frac{n}{2}}+q^{\frac{\alpha+\beta+n+1}{2}};-1,-1,\pm
q^{\frac{1}{2}\alpha},\pm q^{\frac{1}{2}\beta}|q).$ (65)

###### Proof.

Start with expression (58), and the definition of the $q$-Racah polynomials
(4). We choose
$\\{\bar{\alpha},\bar{\beta},\bar{\gamma},\bar{\delta}\\}=\\{\pm
q^{\alpha},\pm q^{\beta},-1,-1\\},$ and therefore $\mu(m)=q^{m+1}+q^{-m}$.
Since $\bar{\delta}=\bar{\gamma}=-1$, the third condition in (5) is
impossible, so we must use either one of the first two conditions. The only
solution is for the positive argument to choose $\alpha=-N-1$ and leave
$\beta$ unrestricted and for the negative argument to chose $\beta=-N-1$ and
to leave $\alpha$ unrestricted. Next, consider (65). We choose
$\\{\bar{\alpha},\bar{\beta},\bar{\gamma},\bar{\delta}\\}=\\{-1,-1,\pm
q^{\alpha},\pm q^{\beta}\\}$, and therefore
$\mu(n)=q^{\alpha+\beta+n+1}+q^{-n}$.

Since $\bar{\alpha}=\bar{\beta}=-1$, the first condition in (5) is impossible,
so we must use either one of the second or third conditions. Again, the only
solution is for the positive argument to choose $\alpha=-N-1$ and leave
$\beta$ unrestricted and for the negative argument to chose $\beta=-N-1$ and
to leave $\alpha$ unrestricted. Since these results are for
$P_{n}^{(\alpha,\beta)}(x|q^{2})$, mapping $q^{2}\mapsto q$, completes the
proof. ∎

######  Corollary 2.11.

If $\beta=-\alpha$ then one has the following duality relations for the
specialized continuous $q$-Jacobi polynomials

$P_{n}^{(\alpha,-\alpha)}(\pm\tfrac{1}{2}(q^{\frac{1}{4}+\frac{m}{2}}+q^{-\frac{1}{4}-\frac{m}{2}})|q),$
(66)

namely

$R_{n}(q^{-\frac{m}{2}}+q^{\frac{m+1}{2}};\pm q^{\frac{1}{2}\alpha},\pm
q^{-\frac{1}{2}\alpha},-1,-1|q)=R_{m}(q^{-\frac{n}{2}}+q^{\frac{n+1}{2}};\pm
q^{\frac{1}{2}\alpha},\pm q^{-\frac{1}{2}\alpha},-1,-1|q).$ (67)

###### Proof.

Replace $\alpha+\beta=0$ in Theorem 2.10 completes the proof. ∎

##  3 The Poisson kernel for continuous $q$-Jacobi polynomials

In general, one may substitute special values of an orthogonal polynomial into
an identity for these polynomials to obtain new specialized identities. We
will now focus on one such application. We now utilize the special values
which we derived in Section 2 within the context of the Poisson kernel for
continuous $q$-Jacobi polynomials. This utilization of special values for
basic hypergeometric orthogonal polynomials is both an application to the
theory of basic hypergeometric orthogonal polynomials and nonterminating basic
hypergeometric functions. The Poisson kernel for an orthogonal polynomial
sequence is a bilinear generating function and is a function of the two
variables, $x=\frac{1}{2}(z+z^{-1})$ and $y=\frac{1}{2}(w+w^{-1})$ (as well as
a power series parameter $|t|<1$ and other parameters involved). Inserting
special values $z$, $w$ (which correspond to $x,y$) in a Poisson kernel
results in the conversion of the bilinear generating function. As we will see
below, replacing either $z$ or $w$ in the Poisson kernel converts it to a
generating function. Furthermore, replacing both $z$ and $w$ in the Poisson
kernel converts it to a transformation formula for an arbitrary argument
(perhaps subject to certain constraints) nonterminating basic hypergeometric
function.

The most general Poisson kernel for Askey–Wilson polynomials ${\sf
K}_{\,t}(x,y):={\sf K}_{\,t}(x,y;{\bf a}|q)$, which is given by

$\displaystyle{\sf
K}_{\,t}(x,y)=\sum_{n=0}^{\infty}\frac{(\frac{abcd}{q},\pm\sqrt{qabcd};q)_{n}\,p_{n}(x;{\bf
a}|q)p_{n}(y;{\bf
a}|q)\,t^{n}}{(q,\pm\sqrt{\frac{abcd}{q}},ab,ac,ad,bc,bd,cd;q)_{n}}$ (68)
$\displaystyle\hskip
28.45274pt=\sum_{n=0}^{\infty}\frac{(\frac{abcd}{q},\pm\sqrt{qabcd},ab,ac,ad;q)_{n}t^{n}}{(q,\pm\sqrt{\frac{abcd}{q}},bc,bd,cd;q)_{n}a^{2n}}r_{n}(x;{\bf
a}|q)r_{n}(y;{\bf a}|q),$ (69)

where we (and as well Gasper & Rahman (1986) [5]) have also used the
normalized version of the Askey–Wilson polynomials (2). For the special case
of the Askey–Wilson polynomials where $ad=bc$, then the Poisson kernel takes a
more simplified form, $K_{t}(x,y):={\sf
K}_{\,t}(x,y;\\{a,b,c,\frac{bc}{a}\\}|q)$ namely

$\displaystyle K_{t}(x,y):=\sum_{n=0}^{\infty}\frac{(\frac{b^{2}c^{2}}{q},\pm
q^{\frac{1}{2}}bc,ab,ac;q)_{n}t^{n}}{(q,\pm
q^{-\frac{1}{2}}bc,\frac{b^{2}c}{a},\frac{bc^{2}}{a};q)_{n}a^{2n}}r_{n}(x;{\bf
a}|q)r_{n}(y;{\bf a}|q).$ (70)

Gasper & Rahman [5, (6.13)] proved a very useful form of this Poisson kernel
for Askey–Wilson polynomials with parameters $(a,b,c,d)$ with the extra
constraint $ad=bc$. This Poisson kernel is given in three terms, each term
given as an infinite sum over a very-well poised and balanced ${}_{10}W_{9}$.
One can see that for the continuous $q$-Jacobi polynomials (7) with the
particular choice of $\\{a,b,c,d\\}$ in (8), the condition $ad=bc$ is
satisfied. Then the Poisson kernel for continuous $q$-Jacobi polynomials is
given by [5, (2.10)]

$K_{t}(x,y):=K_{t}^{(\alpha,\beta)}(x,y|q)=\sum_{n=0}^{\infty}\frac{(q,q^{\alpha+\beta+1},q^{\frac{\alpha+\beta+3}{2}};q)_{n}t^{n}}{(q^{\alpha+1},q^{\beta+1},q^{\frac{\alpha+\beta+1}{2}};q)_{n}q^{(\alpha+\tfrac{1}{2})n}}P_{n}^{(\alpha,\beta)}(x|q)P_{n}^{(\alpha,\beta)}(y|q).$
(71)

Replacing $\\{a,b,c,d\\}$ as in (8) produces the following form for the
Poisson kernel for continuous $q$-Jacobi polynomials.

######  Theorem 3.1.

Let $q\in{{\mathbb{C}}^{\dagger}}$, $\alpha,\beta\in{\mathbb{C}}$, $|t|<1$.
Then the symmetric Poisson kernel for continuous $q$-Jacobi polynomials is
given by

$\displaystyle
K_{t}(x,y)\\!=\\!(1\\!-\\!t^{2})\frac{(-q^{\frac{\alpha+\beta+4}{2}}t;q)_{\infty}}{(-q^{\frac{-\alpha-\beta-2}{2}}t;q)_{\infty}}\sum_{n=0}^{\infty}\frac{(q^{\frac{\alpha+\beta+2}{2}},\pm
q^{\frac{\alpha+\beta+3}{2}},-q^{\frac{1}{2}\beta\\!+\\!\frac{3}{4}}z^{\pm},-q^{\frac{1}{2}\beta\\!+\\!\frac{3}{4}}w^{\pm};q)_{n}q^{n}}{(q,q^{\beta+1},-q^{\frac{1}{2}(\alpha+\beta+\left\\{\begin{subarray}{c}{\scriptsize
2}\\\ {\scriptsize
3}\end{subarray}\right\\})},-q^{\frac{\beta-\alpha+1}{2}},-q^{\frac{\alpha+\beta}{2}+2}t^{\pm};q)_{n}}$
$\displaystyle\hskip
93.89418pt\times\,\mbox{}_{10}W_{9}\\!\left({-q^{\frac{\alpha-\beta-2n-1}{2}}};{q^{-n},-q^{\frac{-\alpha-\beta-2n-1}{2}},q^{-\beta-n},q^{\frac{1}{2}\alpha+\frac{1}{4}}z^{\pm},q^{\frac{1}{2}\alpha+\frac{1}{4}}w^{\pm}};{q,q}\right)$
$\displaystyle\hskip
22.76228pt+\frac{(q^{\alpha+\beta+2},t,-q^{\frac{\alpha-\beta}{2}}t,q^{\frac{1}{2}\alpha+\frac{3}{4}}z^{\pm},q^{\frac{1}{2}\alpha+\frac{1}{4}}w^{\pm},-q^{\frac{1}{2}\beta+\frac{1}{4}}tz^{\pm},-q^{\frac{1}{2}\beta+\frac{3}{4}}tw^{\pm};q)_{\infty}}{(q^{\alpha+1},-q^{\frac{\alpha+\beta}{2}+\left\\{\begin{subarray}{c}{1/2}\\\
{1}\\\
{3/2}\end{subarray}\right\\}},-q^{\frac{\alpha-\beta}{2}},q^{\beta+1}t,-q^{\frac{\alpha+\beta+2}{2}}t^{-1},tz^{\pm}w^{\pm};q)_{\infty}}$
$\displaystyle\hskip
56.9055pt\times\sum_{n=0}^{\infty}\frac{(-t,\pm\sqrt{q}t,q^{\beta+1}t,tz^{\pm}w^{\pm};q)_{n}q^{n}}{(q,-q^{\frac{-\alpha-\beta}{2}}t,qt^{2},-q^{\frac{\alpha-\beta}{2}}t,-q^{\frac{1}{2}\beta+\frac{1}{4}}tz^{\pm},-q^{\frac{1}{2}\beta+\frac{3}{4}}tw^{\pm};q)_{n}}$
$\displaystyle\hskip
93.89418pt\times\,\mbox{}_{10}W_{9}\\!\left({q^{\beta+n}t};{q^{n}t,-q^{\frac{\alpha+\beta+2n}{2}}t,-q^{\frac{\beta-\alpha+2n}{2}}t,-q^{\frac{1}{2}\beta+\frac{3}{4}}z^{\pm},-q^{\frac{1}{2}\beta+\frac{1}{4}}w^{\pm}};{q,q}\right)$
$\displaystyle\hskip
22.76228pt+\frac{(q^{\alpha+\beta+2},t,-q^{\frac{\beta-\alpha}{2}}t,-q^{\frac{1}{2}\beta+\frac{3}{4}}z^{\pm},-q^{\frac{1}{2}\beta+\frac{1}{4}}w^{\pm},q^{\frac{1}{2}{\alpha}+\frac{1}{4}}tz^{\pm},q^{\frac{1}{2}{\alpha}+\frac{3}{4}}tw^{\pm};q)_{\infty}}{(q^{\beta+1},-q^{\frac{\alpha+\beta}{2}+\left\\{\begin{subarray}{c}{1/2}\\\
{1}\\\
{3/2}\end{subarray}\right\\}},-q^{\frac{\beta-\alpha}{2}},q^{\alpha+1}t,-q^{\frac{1}{2}(\alpha+\beta+2)}t^{-1},tz^{\pm}w^{\pm};q)_{\infty}}$
$\displaystyle\hskip
56.9055pt\times\sum_{n=0}^{\infty}\frac{(-t,\pm\sqrt{q}t,q^{\alpha+1}t,tz^{\pm}w^{\pm};q)_{n}q^{n}}{(q,-q^{\frac{-\alpha-\beta}{2}}t,-q^{\frac{\beta-\alpha}{2}}t,qt^{2},q^{\frac{1}{2}\alpha+\frac{1}{4}}tz^{\pm},q^{\frac{1}{2}\alpha+\frac{3}{4}}tw^{\pm};q)_{n}}$
$\displaystyle\hskip
93.89418pt\times\,\mbox{}_{10}W_{9}\\!\left({q^{\alpha+n}t};{q^{n}t,-q^{\frac{\alpha+\beta+2n}{2}},-q^{\frac{\alpha-\beta+2n}{2}}t,q^{\frac{1}{2}\alpha+\frac{3}{4}}z^{\pm},q^{\frac{1}{2}\alpha+\frac{1}{4}}w^{\pm}};{q,q}\right).$
(72)

###### Proof.

This is obtained by making the replacement
$\\{a,b,c,d\\}\mapsto\left\\{q^{\frac{1}{2}\alpha+\frac{1}{4}\left\\{\begin{subarray}{c}{1}\\\
{3}\end{subarray}\right\\}},-q^{\frac{1}{2}\beta+\frac{1}{4}\left\\{\begin{subarray}{c}{1}\\\
{3}\end{subarray}\right\\}}\right\\}$, namely (8), into Gasper & Rahman [5,
(6.13)]. ∎

Note that there is an evident symmetry under the replacement $z\mapsto z^{-1}$
and $w\mapsto w^{-1}$ in (72). Therefore there are 8 possibilities for
substitutions, namely $z,w\in\\{a,b,c,d\\}$. One simple application of the
special values given in Corollary 2.2, is that if you apply these
substitutions to one of the continuous $q$-Jacobi polynomials in the bilinear
generating function given by its Poisson kernel, it is converted to a
generating function. Another easy application of these special values is that
if you apply them to both of the continuous $q$-Jacobi polynomials, then the
Poisson kernel is converted to a transformation formula for an arbitrary
argument basic hypergeometric series. These kind of transformations are fairly
rare in the literature of basic hypergeometric functions. The appearance of
the arbitrary argument in the transformation formula comes from the power
series parameter $t$, which appears in the bilinear generating function.

###  3.1 Generating functions that condense from $K_{t}$

There is no evident symmetry in (72) with respect to $a,b,c,d$, so there are
many choices of replacing with $z$ or $w$ using the special values $a,b,c,d$
given in Corollary 2.2. Upon experimentation, we found that for two particular
choices one obtains some interesting generating functions for continuous
$q$-Jacobi polynomials, which also have interesting $q\to 1^{-}$ limits.

####  3.1.1 The $w=a$ generating function

For the choice $w=a$ we obtain the following interesting generating function
for continuous $q$-Jacobi polynomials.

######  Theorem 3.2.

Let $q\in{{\mathbb{C}}^{\dagger}}$, $x=\frac{1}{2}(z+z^{-1})\in{\mathbb{C}}$,
$\alpha,\beta,t\in{{\mathbb{C}}^{\ast}}$ such that
$|q^{\alpha+\frac{1}{2}}t|<1$. Then

$\displaystyle\sum_{n=0}^{\infty}\frac{(q^{\alpha+\beta+1},q^{\frac{1}{2}(\alpha+\beta+3)};q)_{n}}{(q^{\beta+1},q^{\frac{1}{2}(\alpha+\beta+1)};q)_{n}}t^{n}P_{n}^{(\alpha,\beta)}(x|q)$
$\displaystyle\hskip
14.22636pt=\frac{(-q^{\frac{1}{2}(3\alpha+\beta+5)}t,q^{2\alpha+1}t^{2},-q^{\frac{1}{2}(\alpha+\beta+2)}t,q^{\alpha+\frac{1}{2}\beta+\frac{7}{4}}tz^{\pm};q)_{\infty}}{(q^{2\alpha+2}t^{2},-q^{\alpha+\beta+\frac{5}{2}}t,-q^{\alpha+1}t,q^{\frac{1}{2}\alpha+\frac{1}{4}}tz^{\pm};q)_{\infty}}$
$\displaystyle\hskip
42.67912pt\times\,\mbox{}_{8}W_{7}\\!\left({-q^{\alpha+\beta+\frac{3}{2}}t};{-q^{\alpha+\frac{3}{2}}t,q^{\frac{1}{2}(\beta-\alpha)},q^{\frac{1}{2}(\alpha+\beta+3)},-q^{\frac{1}{2}\beta+\frac{3}{4}}z^{\pm}};{q,-q^{\alpha+\frac{1}{2}}t}\right).$
(73)

###### Proof.

Start with (72) and replacing $w=a$ in Corollary 2.2 and the result follows. ∎

This generating function has a $q\to 1^{-}$ limit given as follows.

######  Theorem 3.3.

Let $x,\alpha,\beta\in{\mathbb{C}}$, $|t|<1$. Then one has the following
generating functions for Jacobi polynomials, namely

$\displaystyle\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_{n}(\frac{1}{2}(\alpha+\beta+3))_{n}}{(\beta+1)_{n}(\frac{1}{2}(\alpha+\beta+1))_{n}}t^{n}P_{n}^{(\alpha,\beta)}(x)$
$\displaystyle\hskip
14.22636pt=\frac{1-t^{2}}{(1+t^{2}-2tx)^{\frac{1}{2}(\alpha+\beta+3)}}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{\frac{1}{2}(\beta-\alpha),\frac{1}{2}(\alpha+\beta+3)}{\beta+1};\frac{-2t(x+1)}{1+t^{2}-2tx}\right)$
$\displaystyle\hskip
14.22636pt=\frac{2^{\frac{1}{2}\beta}(1-t)\Gamma(\beta+1)}{(t(1+x))^{\frac{1}{2}\beta}(1+t^{2}-2tx)^{\frac{1}{2}\alpha+1}}P_{\alpha+1}^{-\beta}\left(\frac{1+t}{\sqrt{1+t^{2}-2tx}}\right),$
(74)

where $P_{\nu}^{\mu}$ is an associated Legendre function of the first kind [4,
(14.3.6)].

###### Proof.

This generating function is obtained by setting
$w=a=q^{\frac{1}{2}\alpha+\frac{1}{4}}$ in (73) and then simplifying the
resulting expression. ∎

As a special case, this generating function leads to the following generating
function for continuous $q$-ultraspherical/Rogers polynomials.

######  Corollary 3.4.

Let $q\in{{\mathbb{C}}^{\dagger}}$, $x=\frac{1}{2}(z+z^{-1})\in{\mathbb{C}}$,
$\beta,t\in{{\mathbb{C}}^{\ast}}$ such that $|t|<1$. Then

$\sum_{n=0}^{\infty}\frac{(q\beta;q)_{n}}{(\beta;q)_{n}}t^{n}C_{n}(x;\beta|q)=(1\\!-\\!\beta
t^{2})\frac{(q\beta tz^{\pm};q)_{\infty}}{(tz^{\pm};q)_{\infty}}$ (75)

###### Proof.

Starting with (73) with $\alpha=\beta$ causes the ${}_{8}W_{7}$ to become
unity. Then using [8, p. 473]

$P_{n}^{(\alpha,\alpha)}(x|q)=\frac{(q^{\alpha+1};q)_{n}}{(q^{2\alpha+1};q)_{n}}q^{(\frac{\alpha}{2}+\frac{1}{4})n}C_{n}(x;q^{\alpha+\frac{1}{2}}|q),$
(76)

completes the proof. ∎

######  Remark 3.5.

The $q\to 1^{-}$ Gegenbauer polynomial limit of the generating function given
by Corollary 3.4 is

$\sum_{n=0}^{\infty}\frac{(1+\beta)_{n}}{(\beta)_{n}}t^{n}C_{n}^{\beta}(x)=\sum_{n=0}^{\infty}\frac{\beta+n}{\beta}t^{n}C_{n}^{\beta}(x)=\frac{1-t^{2}}{(1+t^{2}-2tx)^{\beta+1}}.$
(77)

###  3.2 The $z=d$ generating function

For the choice $z=d$ we obtain the following interesting generating function
for continuous $q$-Jacobi polynomials.

######  Theorem 3.6.

Let $q\in{{\mathbb{C}}^{\dagger}}$, $x=\frac{1}{2}(z+z^{-1})\in{\mathbb{C}}$,
$\alpha,\beta,t\in{{\mathbb{C}}^{\ast}}$ such that
$|q^{\alpha+\frac{1}{2}}t|<1$. Then

$\displaystyle\sum_{n=0}^{\infty}\frac{(q^{\alpha+\beta+1},\pm
q^{\frac{1}{2}(\alpha+\beta+3)};q)_{n}}{(q^{\alpha+1},\pm
q^{\frac{1}{2}(\alpha+\beta+1)};q)_{n}}t^{n}P_{n}^{(\alpha,\beta)}(x|q)$
$\displaystyle\hskip
14.22636pt=\frac{(q^{\alpha+\beta+3}t,q^{\alpha+\beta+2}t^{2};q)_{\infty}}{(t,q^{\alpha+\beta+3}t^{2};q)_{\infty}}\,\mbox{}_{5}\phi_{4}\\!\left(\genfrac{}{}{0.0pt}{}{q^{\frac{1}{2}(\alpha+\beta+2)},\pm
q^{\frac{1}{2}(\alpha+\beta+3)},q^{\frac{1}{2}\alpha+\frac{1}{4}}z^{\pm}}{q^{\alpha+1},-q^{\frac{1}{2}(\alpha+\beta+1)},q^{\alpha+\beta+3}t,\frac{q}{t}};q,q\right)$
$\displaystyle\hskip
22.76228pt+\frac{(q^{\alpha+\beta+2},-q^{\frac{1}{2}(\alpha+\beta+1)}t,-q^{\frac{1}{2}(\alpha+\beta+2)}t,q^{\alpha+1}t,q^{\frac{1}{2}\alpha+\frac{1}{4}}z^{\pm};q)_{\infty}}{(q^{\alpha+1},-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)},\frac{1}{t},q^{\frac{1}{2}\alpha+\frac{1}{4}}tz^{\pm};q)_{\infty}}$
$\displaystyle\hskip
125.19194pt\times\,\mbox{}_{5}\phi_{4}\\!\left(\genfrac{}{}{0.0pt}{}{q^{\frac{1}{2}(\alpha+\beta+2)}t,\pm
q^{\frac{1}{2}(\alpha+\beta+3)}t,q^{\frac{1}{2}\alpha+\frac{1}{4}}tz^{\pm}}{-q^{\frac{1}{2}(\alpha+\beta+1)}t,qt,q^{\alpha+1}t,q^{\alpha+\beta+3}t^{2}};q,q\right).$
(78)

###### Proof.

This generating function is obtained by setting
$z=d=-q^{\frac{1}{2}\beta+\frac{3}{4}}$ in (73), replacing
$y=\frac{1}{2}(w+w^{-1})$ with $x$ and then simplifying the resulting
expression completes the proof. ∎

This generating function for continuous $q$-Jacobi polynomials has a $q\to
1^{-}$ limit given as follows.

######  Theorem 3.7.

Let $x,\alpha,\beta\in{\mathbb{C}}$, $|t|<1$. Then one has the following
generating function for Jacobi polynomials, namely

$\displaystyle\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_{n}(\frac{1}{2}(\alpha+\beta+3))_{n}}{(\alpha+1)_{n}(\frac{1}{2}(\alpha+\beta+1))_{n}}t^{n}P_{n}^{(\alpha,\beta)}(x)$
$\displaystyle\hskip
56.9055pt=\frac{1-t^{2}}{(1-t)^{\alpha+\beta+3}}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{\frac{1}{2}(\alpha+\beta+2),\frac{1}{2}(\alpha+\beta+3)}{\alpha+1};\frac{2t(y-1)}{(1-t)^{2}}\right)$
$\displaystyle\hskip
56.9055pt=\frac{2^{\frac{1}{2}\alpha}(1+t)\Gamma(\alpha+1)}{(t(y-1))^{\frac{1}{2}\alpha}(1+t^{2}-2ty)^{\frac{1}{2}\beta+1}}P_{\beta+1}^{-\alpha}\left(\frac{1-t}{\sqrt{1+t^{2}-2ty}}\right),$
(79)

where $P_{\nu}^{\mu}$ is an associated Legendre function of the first kind [4,
(14.3.6)].

Starting from (78), using the symmetric limit $\alpha=\beta$, we can produce a
new generating function for continuous $q$-ultraspherical polynomials.

######  Corollary 3.8.

Let $q\in{{\mathbb{C}}^{\dagger}}$, $x=\frac{1}{2}(z+z^{-1})\in{\mathbb{C}}$,
$\beta,t\in{{\mathbb{C}}^{\ast}}$ such that $|t|<1$. Then

$\displaystyle\sum_{n=0}^{\infty}\frac{(\pm
q\beta;q)_{n}}{(\pm\beta;q)_{n}}t^{n}C_{n}(x;\beta|q)=\frac{(q\beta,-q\beta
tz^{\pm},\beta t^{2};q)_{\infty}}{(-\beta,tz^{\pm},-q\beta
t^{2};q)_{\infty}}\,\mbox{}_{8}W_{7}\\!\left({-\beta t^{2}};{-\frac{1}{q},\pm
q\sqrt{\beta}t,tz^{\pm}};{q,q\beta}\right).$ (80)

###### Proof.

Starting with (78) with $\alpha=\beta$ causes the sum of two
${}_{5}\phi_{4}(q,q)$s to become a sum of two ${}_{4}\phi_{3}(q,q)$s which
then naturally transforms to an ${}_{8}W_{7}$ using Bailey’s transformation
[4, (17.9.16)]. Then using (76) completes the proof. ∎

####  3.2.1 Arbitrary argument transformations that arise from Gasper &
Rahman’s Poisson kernel for continuous $q$-Jacobi polynomials

As mentioned just below (72), if one makes the double replacement of $z$ and
$w$ in the Poisson kernel, one obtains an arbitrary argument transformation
formula. Because of limitations on space in the current manuscript, we will
only treat a single example. Note however, that if one considers the Poisson
kernel for continuous $q$-Jacobi polynomials and then one takes the special
values (41)–(44) for $z$ and $w$ simultaneously, then there are $4\times 4=16$
combinations. For each case, the Poisson kernel reduces to a single
nonterminating basic hypergeometric series with an arbitrary argument. The
cases $z=w$ correspond to 4 unique transformations and the 12 off-diagonal
combinations combine into pairs, with 6 paired transformations for a single
nonterminating basic hypergeometric series. Below we present the results for
the computation of these transformations. For the $z=w=a$ case, then the
Poisson kernel produces a well-poised ${}_{3}\phi_{2}$ with arbitrary argument
$z$. This is a well-poised nonterminating ${}_{3}\phi_{2}$ with arbitrary
argument $z$ expressed as a sum of two ${}_{4}\phi_{3}$s with argument $q$ or
a nonterminating ${}_{8}W_{7}$.

######  Theorem 3.9.

Let $a,b,c\in{\mathbb{C}}$ such that $|a|,|z|<1$,
$q\in{{\mathbb{C}}^{\dagger}}$. Then the following well-poised
${}_{3}\phi_{2}$ is given by

$\displaystyle\,\mbox{}_{3}\phi_{2}\\!\left(\genfrac{}{}{0.0pt}{}{ab,\sqrt{q}a,q\sqrt{ab}}{\sqrt{q}b,\sqrt{ab}};q,z\right)=\frac{(a^{2}z^{2},-\sqrt{q^{3}a^{3}b}z;q)_{\infty}}{(qa^{2}z^{2},-\sqrt{a/(qb)}z;q)_{\infty}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{q\sqrt{ab},\pm\sqrt{qab},-\sqrt{\frac{qb}{a}}}{\sqrt{q}b,-\sqrt{q^{3}a^{3}b}z,-\sqrt{\frac{q^{3}b}{a}}z^{-1}};q,q\right)$
$\displaystyle\hskip
56.9055pt+\frac{(qab,\sqrt{q}az,-\sqrt{qb/a},-\sqrt{ab}z;q)_{\infty}}{(\sqrt{q}b,-q\sqrt{ab},z,-\sqrt{qb/a}z^{-1};q)_{\infty}}\,\mbox{}_{4}\phi_{3}\\!\left(\genfrac{}{}{0.0pt}{}{z,\pm
az,-\sqrt{q}az}{qa^{2}z^{2},-\sqrt{\frac{qa}{b}}z,-\sqrt{ab}z};q,q\right)$
(81) $\displaystyle\hskip
14.22636pt=\frac{(a^{2}z^{2},q\sqrt{ab}z,-\sqrt{qab}z,q\sqrt{a^{3}b}z,-\sqrt{q^{3}a^{3}b}z;q)_{\infty}}{(qa^{2}z^{2},z,az,-\sqrt{q}az,-\sqrt{q^{3}}abz;q)_{\infty}}$
$\displaystyle\hskip
56.9055pt\times\,\mbox{}_{8}W_{7}\\!\left({-\sqrt{q}abz};{q\sqrt{ab},-\sqrt{qab},\sqrt{\frac{b}{a}},-\sqrt{\frac{qb}{a}},-qaz};{q,-az}\right).$
(82)

###### Proof.

Start with the Poisson kernel for continuous $q$-Jacobi polynomials and
substitute $z=w=a$. This converts the infinite series into a single
nonterminating ${}_{3}\phi_{2}$. If one takes $q^{\alpha+\frac{1}{2}}\mapsto
a$, $q^{\beta+\frac{1}{2}}\mapsto b$, then one arrives at a nonterminating
transformation which is a sum of two ${}_{4}\phi_{3}$s with argument $q$.
Using Bailey’s transformation [4, (17.9.16)] we convert it to a nonterminating
${}_{8}W_{7}$. This completes the proof. ∎

### Acknowledgements

We would like to thank Tom Koornwinder, Mourad E. H. Ismail and George Gasper
for valuable discussions.

## References

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