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KLSadd.tex
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%until 185
\documentclass[twoside,11pt]{article}
\usepackage[pdftex]{hyperref}
\hypersetup{
colorlinks = true,
allcolors = {red},
}
\usepackage{color}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{xparse}
\usepackage{cite}
\renewcommand\citeform[1]{K#1}
\setlength{\textwidth}{16cm}
\setlength{\textheight}{21cm}
\setlength{\topmargin}{0cm}
\setlength{\oddsidemargin}{0.2mm}
\setlength{\evensidemargin}{0.2mm}
\newcommand\sa{\smallskipamount}
\newcommand\sLP{\\[\sa]}
\newcommand\sPP{\\[\sa]\indent}
\newcommand\ba{\bigskipamount}
\newcommand\bLP{\\[\ba]}
\newcommand\CC{\mathbb{C}}
\newcommand\RR{\mathbb{R}}
\newcommand\ZZ{\mathbb{Z}}
\newcommand\al\alpha
\newcommand\be\beta
\newcommand\ga\gamma
\newcommand\de\delta
\newcommand\tha\theta
\newcommand\la\lambda
\newcommand\om\omega
\newcommand\Ga{\Gamma}
\newcommand\half{\frac12}
\newcommand\thalf{\tfrac12}
\newcommand\iy\infty
\newcommand\wt{\widetilde}
\newcommand\const{{\rm const.}\,}
\newcommand\Zpos{\ZZ_{>0}}
\newcommand\Znonneg{\ZZ_{\ge0}}
\newcommand{\hyp}[5]{\,\mbox{}_{#1}F_{#2}\!\left(
\genfrac{}{}{0pt}{}{#3}{#4};#5\right)}
\newcommand{\qhyp}[5]{\,\mbox{}_{#1}\phi_{#2}\!\left(
\genfrac{}{}{0pt}{}{#3}{#4};#5\right)}
\newcommand\LHS{left-hand side}
\newcommand\RHS{right-hand side}
\renewcommand\Re{{\rm Re}\,}
\renewcommand\Im{{\rm Im}\,}
\NewDocumentCommand\mycite{m g}{%
\IfNoValueTF{#2}
{[\hyperlink{#1}{#1}]}
{[\hyperlink{#1}{#1}, #2]}%
}
\newcommand\mybibitem[1]{\bibitem[#1]{#1}\hypertarget{#1}{}}
%\NewDocumentCommand{\myciteKLS}{m g}{%
% \IfNoValueTF{#2}
% {\hyperlink{KLS#1}{[#1]}}
% {\hyperlink{KLS#1}{[#1, #2]}}%
%}
\NewDocumentCommand{\myciteKLS}{m g}{%
\IfNoValueTF{#2}
{[\hyperlink{KLS#1}{#1}]}
{[\hyperlink{KLS#1}{#1}, #2]}%
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\newcommand\mybibitemKLS[1]{\bibitem[#1]{#1}\hypertarget{KLS#1}{}}
\begin{document}
\title{Additions to the formula lists in
``Hypergeometric orthogonal polynomials and their $q$-analogues''
by Koekoek, Lesky and Swarttouw}
\author{Tom H. Koornwinder}
\date{June 19, 2015}
\maketitle
\begin{abstract}
This report gives a rather arbitrary choice of formulas for
($q$-)hypergeometric orthogonal polynomials which the author missed
while consulting Chapters 9 and 14 in the book
``Hypergeometric orthogonal polynomials and their $q$-analogues''
by Koekoek, Lesky and Swarttouw. The systematics of these chapters will be followed
here, in particular for the numbering of subsections and of references.
\end{abstract}
%
\subsection*{Introduction}
\label{sec_intro}
This report contains some formulas about ($q$-)hypergeometric
orthogonal polynomials which I missed but wanted to use
while consulting Chapters 9 and 14 in the book \mycite{KLS}:
\sLP
R. Koekoek, P.~A. Lesky and R.~F. Swarttouw,
{\em Hypergeometric orthogonal polynomials and their $q$-analogues},
Springer-Verlag, 2010.
\sLP
These chapters form together the (slightly extended) successor of the report
\sLP
R.~Koekoek and R.~F. Swarttouw,
{\em The Askey-scheme of hypergeometric orthogonal
polynomials and its $q$-analogue},
Report 98-17, Faculty of Technical Mathematics and Informatics,
Delft University of Technology, 1998;
\url{http://aw.twi.tudelft.nl/~koekoek/askey/}.
\sPP
Certainly these chapters give complete lists of formulas of special type, for instance
orthogonality relations and three-term recurrence relations. But outside these narrow
categories there are many other
formulas for ($q$-)orthogonal polynomials which one wants to have available.
Often one can find the desired formula in one of the
\hyperref[sec_ref1]{standard references} listed at the end of this report.
Sometimes it is only available in a journal or a less common monograph.
Just for my own comfort, I have brought together some of these formulas.
This will possibly also be helpful for some other users.
Usually, any type of formula I give for a special class of polynomials, will suggest
a similar formula for many other classes, but I have not aimed at completeness
by filling in a formula of such type at all places. The resulting choice of formulas is
rather arbitrary, just depending on the formulas which I happened to need or which raised my interest.
For each formula I give a suitable reference or I sketch a
proof.
It is my intention to gradually extend this collection of formulas.
%
\subsection*{Conventions}
\label{sec_conv}
The (x.y) and (x.y.z) type subsection numbers, the
(x.y.z) type formula numbers, and the [x] type citation numbers
refer to \mycite{KLS}.
The (x) type formula numbers refer to this manuscript and the [Kx] type citation numbers refer to citations which are not in \mycite{KLS}.
Some standard references like \mycite{DLMF}
are given by special acronyms.
$N$ is always a positive integer. Always assume $n$ to be a nonnegative
integer or, if $N$ is present, to be in $\{0,1,\ldots,N\}$.
Throughout assume $0<q<1$.
For each family the coefficient of the term of highest degree of the
orthogonal polynomial of degree $n$ can be found in \mycite{KLS} as the
coefficient of $p_n(x)$ in the formula after the main formula under
the heading ``Normalized Recurrence Relation". If that main formula is numbered
as (x.y.z) then I will refer to the second formula as (x.y.zb).
In the notation of $q$-hypergeometric orthogonal polynomials we
will follow the convention that the parameter list and $q$ are separated
by `$\,|\,$' in the case of a $q$-quadratic lattice (for instance
\hyperref[sec14.1]{Askey-Wilson})
and by `;' in the case of a $q$-linear lattice (for instance
\hyperref[sec14.5]{big $q$-Jacobi}). This convention is mostly followed
in \mycite{KLS}, but not everywhere, see for instance
\hyperref[sec14.20]{little $q$-Laguerre / Wall}.
%
\subsection*{Acknowledgement}
Many thanks to Howard Cohl for having called my attention so often to typos and
inconsistencies.
%
\newpage
\subsection*{Contents}
\hyperref[sec_intro]{Introduction}\\
\hyperref[sec_conv]{Conventions}\\
\hyperref[sec_general]{Generalities}
\sLP
\hyperref[sec9.1]{9.1 Wilson}\\
\hyperref[sec9.2]{9.2 Racah}\\
\hyperref[sec9.3]{9.3 Continuous dual Hahn}\\
\hyperref[sec9.4]{9.4 Continuous Hahn}\\
\hyperref[sec9.5]{9.5 Hahn}\\
\hyperref[sec9.6]{9.6 Dual Hahn}\\
\hyperref[sec9.7]{9.7 Meixner-Pollaczek}\\
\hyperref[sec9.8]{9.8 Jacobi}
\hyperref[sec9.8.1]{9.8.1 Gegenbauer / Ultraspherical}
\hyperref[sec9.8.2]{9.8.2 Chebyshev}\\
\hyperref[sec9.9]{9.9 Pseudo Jacobi (or Routh-Romanovski)}\\
\hyperref[sec9.10]{9.10 Meixner}\\
\hyperref[sec9.11]{9.11 Krawtchouk}\\
\hyperref[sec9.12]{9.12 Laguerre}\\
\hyperref[sec9.14]{9.14 Charlier}\\
\hyperref[sec9.15]{9.15 Hermite}
\sLP
\hyperref[sec14.1]{14.1 Askey-Wilson}\\
\hyperref[sec14.2]{14.2 $q$-Racah}\\
\hyperref[sec14.3]{14.3 Continuous dual $q$-Hahn}\\
\hyperref[sec14.4]{14.4 Continuous $q$-Hahn}\\
\hyperref[sec14.5]{14.5 Big $q$-Jacobi}\\
\hyperref[sec14.7]{14.7 Dual $q$-Hahn}\\
\hyperref[sec14.8]{14.8 Al-Salam-Chihara}\\
\hyperref[sec14.9]{14.9 $q$-Meixner-Pollaczek}\\
\hyperref[sec14.10]{14.10 Continuous $q$-Jacobi}
\hyperref[sec14.10.1]{14.10.1 Continuous $q$-ultraspherical / Rogers}\\
\hyperref[sec14.11]{14.11 Big $q$-Laguerre}\\
\hyperref[sec14.12]{14.12 Little $q$-Jacobi}\\
\hyperref[sec14.14]{14.14 Quantum $q$-Krawtchouk}\\
\hyperref[sec14.16]{14.16 Affine $q$-Krawtchouk}\\
\hyperref[sec14.17]{14.17 Dual $q$-Krawtchouk}\\
\hyperref[sec14.20]{14.20 Little $q$-Laguerre / Wall}\\
\hyperref[sec14.21]{14.21 $q$-Laguerre}\\
\hyperref[sec14.27]{14.27 Stieltjes-Wigert}\\
\hyperref[sec14.28]{14.28 Discrete $q$-Hermite I}\\
\hyperref[sec14.29]{14.29 Discrete $q$-Hermite II}
\sLP
\hyperref[sec_ref1]{Standard references}\\
\hyperref[sec_ref2]{References from [KLS]}\\
\hyperref[sec_ref3]{Other references}
%
\newpage
%
\subsection*{Generalities}
\label{sec_general}
\paragraph{Criteria for uniqueness of orthogonality measure}
According to Shohat \& Tamarkin \cite[p.50]{K6}
orthonormal polynomials $p_n$ have a unique orthogonality measure (up to positive
constant factor) if
for some $z\in\CC$ we have
\begin{equation}
\sum_{n=0}^\iy |p_n(z)|^2 = \iy.
\label{90}
\end{equation}
Also (see Shohat \& Tamarkin \cite[p.59]{K6}),
monic orthogonal polynomials $p_n$ with three-term recurrence relation
$x p_n(x) = p_{n+1}(x)+B_n p_n(x)+C_n p_{n-1}(x)$
($C_n$ necessarily positive)
have a unique orthogonality measure if
\begin{equation}
\sum_{n=1}^\iy (C_n)^{-1/2}=\iy.
\label{93}
\end{equation}
Furthermore, if orthogonal polynomials have an orthogonality measure with
bounded support, then this is unique (see Chihara \myciteKLS{146}).
%
\paragraph{Even orthogonality measure}
If $\{p_n\}$ is a system of orthogonal polynomials with respect to an even
orthogonality measure which satisfies the three-term recurrence relation
\begin{equation*}
x p_n(x)=A_n p_{n+1}(x)+C_n p_{n-1}(x)
\end{equation*}
then
\begin{equation}
\frac{p_{2n}(0)}{p_{2n-2}(0)}=-\,\frac{C_{2n-1}}{A_{2n-1}}\,.
\label{1}
\end{equation}
%
\paragraph{Appell's bivariate hypergeometric function $F_4$}
This is defined by
\begin{equation}
F_4(a,b;c,c';x,y):=\sum_{m,n=0}^\iy\frac{(a)_{m+n}(b)_{m+n}}{(c)_m(c')_n\,m!\,n!}\,
x^my^n\qquad(|x|^\half+|y|^\half<1),
\label{62}
\end{equation}
see \mycite{HTF1}{5.7(9), 5.7(44)} or \mycite{DLMF}{(16.13.4)}.
There is the reduction formula
\begin{equation*}
F_4\left(a,b;b,b;\frac{-x}{(1-x)(1-y)},\frac{-y}{(1-x)(1-y)}\right)=
(1-x)^a(1-y)^a\,\hyp21{a,1+a-b}b{xy},
\end{equation*}
see \mycite{HTF1}{5.10(7)}. When combined with the quadratic transformation
\mycite{HTF1}{2.11(34)} (here $a-b-1$ should be replaced by $a-b+1$),
see also \mycite{DLMF}{(15.8.15)}, this yields
\begin{multline*}
F_4\left(a,b;b,b;\frac{-x}{(1-x)(1-y)},\frac{-y}{(1-x)(1-y)}\right)\\
=\left(\frac{(1-x)(1-y)}{1+xy}\right)^a\,
\hyp21{\thalf a,\thalf(a+1)}b{\frac{4xy}{(1+xy)^2}}.
\end{multline*}
This can be rewritten as
\begin{equation}
F_4(a,b;b,b;x,y)=(1-x-y)^{-a}\,\hyp21{\thalf a,\thalf(a+1)}b
{\frac{4xy}{(1-x-y)^2}}.
\label{63}
\end{equation}
Note that, if $x,y\ge0$ and $x^\half+y^\half<1$, then
$1-x-y>0$ and $0\le\frac{4xy}{(1-x-y)^2}<1$.
%
\paragraph{$q$-Hypergeometric series of base $q^{-1}$}
By \mycite{GR}{Exercise 1.4(i)}:
\begin{equation}
\qhyp rs{a_1,\ldots,a_r}{b_1,\ldots b_s}{q^{-1},z}
=\qhyp{s+1}s{a_1^{-1},\ldots a_r^{-1},0,\ldots,0}
{b_1^{-1},\ldots,b_s^{-1}}{q,\frac{qa_1\ldots a_rz}{b_1\ldots b_s}}
\label{154}
\end{equation}
for $r\le s+1$, $a_1,\ldots,a_r,b_1,\ldots,b_s\ne0$.
In the non-terminating case, for $0<q<1$, there is convergence if
$|z|<b_1\ldots b_s/(qa_1\ldots a_r)$\,.
%
\paragraph{A transformation of a terminating ${}_2\phi_1$}
By \mycite{GR}{Exercise 1.15(i)} we have
\begin{equation}
\qhyp21{q^{-n},b}c{q,z}=(bz/(cq);q^{-1})_n\,
\qhyp32{q^{-n},c/b,0}{c,cq/(bz)}{q,q}.
\label{151}
\end{equation}
%
\paragraph{Very-well-poised $q$-hypergeometric series}
The notation of \mycite{GR}{(2.1.11)} will be followed:
\begin{equation}
{}_{r+1}W_r(a_1;a_4,a_5,\ldots,a_{r+1};q,z):=
\qhyp{r+1}r{a_1,qa_1^\half,-qa_1^\half,a_4,\ldots,a_{r+1}}
{a_1^\half,-a_1^\half,qa_1/a_4,\ldots,qa_1/a_{r+1}}{q,z}.
\label{111}
\end{equation}
%
\paragraph{Theta function}
The notation of \mycite{GR}{(11.2.1)} will be followed:
\begin{equation}
\tha(x;q):=(x,q/x;q)_\iy,\qquad
\tha(x_1,\ldots,x_m;q):=\tha(x_1;q)\ldots\tha(x_m;q).
\label{117}
\end{equation}
%
\subsection*{9.1 Wilson}
\label{sec9.1}
%
\paragraph{Symmetry}
The Wilson polynomial $W_n(y;a,b,c,d)$ is symmetric
in $a,b,c,d$.
\\
This follows from the orthogonality relation (9.1.2)
together with the value of its coefficient of $y^n$ given in (9.1.5b).
Alternatively, combine (9.1.1) with \mycite{AAR}{Theorem 3.1.1}.\\
As a consequence, it is sufficient to give generating function (9.1.12). Then the generating
functions (9.1.13), (9.1.14) will follow by symmetry in the parameters.
%
\paragraph{Hypergeometric representation}
In addition to (9.1.1) we have (see \myciteKLS{513}{(2.2)}):
\begin{multline}
W_n(x^2;a,b,c,d)
=\frac{(a-ix)_n (b-ix)_n (c-ix)_n (d-ix)_n}{(-2ix)_n}\\
\times\hyp76{2ix-n,ix-\thalf n+1,a+ix,b+ix,c+ix,d+ix,-n}
{ix-\thalf n,1-n-a+ix,1-n-b+ix,1-n-c+ix,1-n-d+ix}1.
\label{112}
\end{multline}
The symmetry in $a,b,c,d$ is clear from \eqref{112}.
%
\paragraph{Special value}
\begin{equation}
W_n(-a^2;a,b,c,d)=(a+b)_n(a+c)_n(a+d)_n\,,
\label{91}
\end{equation}
and similarly for arguments $-b^2$, $-c^2$ and
$-d^2$ by symmetry of $W_n$ in $a,b,c,d$.
%
\paragraph{Uniqueness of orthogonality measure}
Under the assumptions on $a,b,c,d$ for (9.1.2) or (9.1.3) the orthogonality
measure is unique up to constant factor.
For the proof assume without
loss of generality (by the symmetry in $a,b,c,d$) that $\Re a\ge0$.
Write the \RHS\ of (9.1.2) or (9.1.3) as $h_n\de_{m,n}$.
Observe from (9.1.2) and \eqref{91} that
\[
\frac{|W_n(-a^2;a,b,c,d)|^2}{h_n} = O(n^{4\Re a-1})\quad\hbox{as $n\to\iy$.}
\]
Therefore \eqref{90} holds, from which the uniqueness of the orthogonality
measure follows.
By a similar, but necessarily more complicated argument Ismail et al.\
\myciteKLS{281}{Section 3} proved the uniqueness of orthogonality measure for
associated Wilson polynomials.
%
\subsection*{9.2 Racah}
\label{sec9.2}
\paragraph{Racah in terms of Wilson}
In the Remark on p.196 Racah polynomials are expressed in terms of
Wilson polynomials. This can be equivalently written as
\begin{multline}
R_n\big(x(x-N+\de);\al,\be,-N-1,\de\big)\\
=\frac{W_n\big(-(x+\thalf(\de-N))^2;\thalf(\de-N),\al+1-\thalf(\de-N),
\be+\thalf(\de+N)+1,-\half(\de+N)\big)}
{(\al+1)_n (\be+\de+1)_n (-N)_n}\,.
\label{146}
\end{multline}
%
\subsection*{9.3 Continuous dual Hahn}
\label{sec9.3}
%
\paragraph{Symmetry}
The continuous dual Hahn polynomial $S_n(y;a,b,c)$ is symmetric
in $a,b,c$.\\
This follows from the orthogonality relation (9.3.2)
together with the value of its coefficient of $y^n$ given in (9.3.5b).
Alternatively, combine (9.3.1) with \mycite{AAR}{Corollary 3.3.5}.\\
As a consequence, it is sufficient to give generating function (9.3.12). Then the generating
functions (9.3.13), (9.3.14) will follow by symmetry in the parameters.
%
\paragraph{Special value}
\begin{equation}
S_n(-a^2;a,b,c)=(a+b)_n(a+c)_n\,,
\label{92}
\end{equation}
and similarly for arguments $-b^2$ and $-c^2$ by symmetry of $S_n$ in $a,b,c$.
%
\paragraph{Uniqueness of orthogonality measure}
Under the assumptions on $a,b,c$ for (9.3.2) or (9.3.3) the orthogonality
measure is unique up to constant factor.
For the proof assume without
loss of generality (by the symmetry in $a,b,c,d$) that $\Re a\geq0$.
Write the \RHS\ of (9.3.2) or (9.3.3) as $h_n\de_{m,n}$.
Observe from (9.3.2) and \eqref{92} that
\[
\frac{|S_n(-a^2;a,b,c)|^2}{h_n} = O(n^{2\Re a-1})\quad
\hbox{as $n\to\iy$.}
\]
Therefore \eqref{90} holds, from which the uniqueness of the orthogonality
measure follows.
%
\subsection*{9.4 Continuous Hahn}
\label{sec9.4}
%
\paragraph{Orthogonality relation and symmetry}
The orthogonality relation (9.4.2) holds under the more general assumption that
$\Re(a,b,c,d)>0$ and $(c,d)=(\overline a,\overline b)$ or $(\overline b,\overline a)$.\\
Thus, under these assumptions, the continuous Hahn polynomial
$p_n(x;a,b,c,d)$
is symmetric in $a,b$ and in $c,d$.
This follows from the orthogonality relation (9.4.2)
together with the value of its coefficient of $x^n$ given in (9.4.4b).\\
As a consequence, it is sufficient to give generating function (9.4.11). Then the generating
function (9.4.12) will follow by symmetry in the parameters.
%
\paragraph{Uniqueness of orthogonality measure}
The coefficient of $p_{n-1}(x)$ in (9.4.4) behaves as $O(n^2)$ as $n\to\iy$.
Hence \eqref{93} holds, by which the orthogonality measure is unique.
%
\paragraph{Special cases}
In the following special case there is a reduction to
Meixner-Pollaczek:
\begin{equation}
p_n(x;a,a+\thalf,a,a+\thalf)=
\frac{(2a)_n (2a+\thalf)_n}{(4a)_n}\,P_n^{(2a)}(2x;\thalf\pi).
\end{equation}
See \myciteKLS{342}{(2.6)} (note that in \myciteKLS{342}{(2.3)} the
Meixner-Pollaczek polynonmials are defined different from (9.7.1),
without a constant factor in front).
For $0<a<1$ the continuous Hahn polynomials $p_n(x;a,1-a,a,1-a)$
are orthogonal on $(-\iy,\iy)$ with respect to the weight function
$\big(\cosh(2\pi x)-\cos(2\pi a)\big)^{-1}$
(by straightforward computation from (9.4.2)).
For $a=\tfrac14$ the two special cases coincide:
Meixner-Pollaczek with weight function $\big(\cosh(2\pi x)\big)^{-1}$.
%
\subsection*{9.5 Hahn}
\label{sec9.5}
%
\paragraph{Special values}
\begin{equation}
Q_n(0;\al,\be,N)=1,\quad
Q_n(N;\al,\be,N)=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,.
\label{95}
\end{equation}
Use (9.5.1) and compare with (9.8.1) and \eqref{50}.
From (9.5.3) and \eqref{1} it follows that
\begin{equation}
Q_{2n}(N;\al,\al,2N)=\frac{(\thalf)_n(N+\al+1)_n}{(-N+\thalf)_n(\al+1)_n}\,.
\label{30}
\end{equation}
From (9.5.1) and \mycite{DLMF}{(15.4.24)} it follows that
\begin{equation}
Q_N(x;\al,\be,N)=\frac{(-N-\be)_x}{(\al+1)_x}\qquad(x=0,1,\ldots,N).
\label{44}
\end{equation}
%
\paragraph{Symmetries}
By the orthogonality relation (9.5.2):
\begin{equation}
\frac{Q_n(N-x;\al,\be,N)}{Q_n(N;\al,\be,N)}=Q_n(x;\be,\al,N),
\label{96}
\end{equation}
It follows from \eqref{97} and \eqref{45} that
\begin{equation}
\frac{Q_{N-n}(x;\al,\be,N)}{Q_N(x;\al,\be,N)}
=Q_n(x;-N-\be-1,-N-\al-1,N)
\qquad(x=0,1,\ldots,N).
\label{100}
\end{equation}
%
\paragraph{Duality}
The Remark on p.208 gives the duality between Hahn and dual Hahn polynomials:
%
\begin{equation}
Q_n(x;\al,\be,N)=R_x(n(n+\al+\be+1);\al,\be,N)\quad(n,x\in\{0,1,\ldots N\}).
\label{45}
\end{equation}
%
\subsection*{9.6 Dual Hahn}
\label{sec9.6}
%
\paragraph{Special values}
By \eqref{44} and \eqref{45} we have
\begin{equation}
R_n(N(N+\ga+\de+1);\ga,\de,N)=\frac{(-N-\de)_n}{(\ga+1)_n}\,.
\label{47}
\end{equation}
It follows from \eqref{95} and \eqref{45} that
\begin{equation}
R_N(x(x+\ga+\de+1);\ga,\de,N)
=\frac{(-1)^x(\de+1)_x}{(\ga+1)_x}\qquad(x=0,1,\ldots,N).
\label{101}
\end{equation}
%
\paragraph{Symmetries}
Write the weight in (9.6.2) as
\begin{equation}
w_x(\al,\be,N):=N!\,\frac{2x+\ga+\de+1}{(x+\ga+\de+1)_{N+1}}\,
\frac{(\ga+1)_x}{(\de+1)_x}\,\binom Nx.
\label{98}
\end{equation}
Then
\begin{equation}
(\de+1)_N\,w_{N-x}(\ga,\de,N)=
(-\ga-N)_N\,w_x(-\de-N-1,-\ga-N-1,N).
\label{99}
\end{equation}
Hence, by (9.6.2),
\begin{equation}
\frac{R_n((N-x)(N-x+\ga+\de+1);\ga,\de,N)}{R_n(N(N+\ga+\de+1);\ga,\de,N)}
=R_n(x(x-2N-\ga-\de-1);-N-\de-1,-N-\ga-1,N).
\label{97}
\end{equation}
Alternatively, \eqref{97} follows from (9.6.1) and
\mycite{DLMF}{(16.4.11)}.
It follows from \eqref{96} and \eqref{45} that
\begin{equation}
\frac{R_{N-n}(x(x+\ga+\de+1);\ga,\de,N)}
{R_N(x(x+\ga+\de+1);\ga,\de,N)}
=R_n(x(x+\ga+\de+1);\de,\ga,N)\qquad(x=0,1,\ldots,N).
\label{102}
\end{equation}
%
\paragraph{Re: (9.6.11).}
The generating function (9.6.11) can be written in a more conceptual way as
\begin{equation}
(1-t)^x\,\hyp21{x-N,x+\ga+1}{-\de-N}t=\frac{N!}{(\de+1)_N}\,
\sum_{n=0}^N \om_n\,R_n(\la(x);\ga,\de,N)\,t^n,
\label{2}
\end{equation}
where
\begin{equation}
\om_n:=\binom{\ga+n}n \binom{\de+N-n}{N-n},
\label{3}
\end{equation}
i.e., the denominator on the \RHS\ of (9.6.2).
By the duality between Hahn polynomials and dual Hahn polynomials (see \eqref{45}) the above generating function can be rewritten in
terms of Hahn polynomials:
\begin{equation}
(1-t)^n\,\hyp21{n-N,n+\al+1}{-\be-N}t=\frac{N!}{(\be+1)_N}\,
\sum_{x=0}^N w_x\,Q_n(x;\al,\be,N)\,t^x,
\label{4}
\end{equation}
where
\begin{equation}
w_x:=\binom{\al+x}x \binom{\be+N-x}{N-x},
\label{5}
\end{equation}
i.e., the weight occurring in the orthogonality relation (9.5.2)
for Hahn polynomials.
\paragraph{Re: (9.6.15).}
There should be a closing bracket before the equality sign.
%
\subsection*{9.7 Meixner-Pollaczek}
\label{sec9.7}
%
\paragraph{Uniqueness of orthogonality measure}
The coefficient of $p_{n-1}(x)$ in (9.7.4) behaves as $O(n^2)$ as $n\to\iy$.
Hence \eqref{93} holds, by which the orthogonality measure is unique.
%
\subsection*{9.8 Jacobi}
\label{sec9.8}
%
\paragraph{Orthogonality relation}
Write the \RHS\ of (9.8.2) as $h_n\,\de_{m,n}$. Then
\begin{equation}
\begin{split}
&\frac{h_n}{h_0}=
\frac{n+\al+\be+1}{2n+\al+\be+1}\,
\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad
h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(\be+1)}{\Ga(\al+\be+2)}\,,\sLP
&\frac{h_n}{h_0\,(P_n^{(\al,\be)}(1))^2}=
\frac{n+\al+\be+1}{2n+\al+\be+1}\,
\frac{(\be+1)_n\,n!}{(\al+1)_n\,(\al+\be+2)_n}\,.
\end{split}
\label{60}
\end{equation}
In (9.8.3) the numerator factor $\Ga(n+\al+\be+1)$ in the last line should be
$\Ga(\be+1)$. When thus corrected, (9.8.3) can be rewritten as:
\begin{equation}
\begin{split}
&\int_1^\iy P_m^{(\al,\be)}(x)\,P_n^{(\al,\be)}(x)\,(x-1)^\al (x+1)^\be\,dx=h_n\,\de_{m,n}\,,\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad-1-\be>\al>-1,\quad m,n<-\thalf(\al+\be+1),\\
&\frac{h_n}{h_0}=
\frac{n+\al+\be+1}{2n+\al+\be+1}\,
\frac{(\al+1)_n(\be+1)_n}{(\al+\be+2)_n\,n!}\,,\quad
h_0=\frac{2^{\al+\be+1}\Ga(\al+1)\Ga(-\al-\be-1)}{\Ga(-\be)}\,.
\end{split}
\label{122}
\end{equation}
%
\paragraph{Symmetry}
\begin{equation}
P_n^{(\al,\be)}(-x)=(-1)^n\,P_n^{(\be,\al)}(x).
\label{48}
\end{equation}
Use (9.8.2) and (9.8.5b) or see \mycite{DLMF}{Table 18.6.1}.
%
\paragraph{Special values}
\begin{equation}
P_n^{(\al,\be)}(1)=\frac{(\al+1)_n}{n!}\,,\quad
P_n^{(\al,\be)}(-1)=\frac{(-1)^n(\be+1)_n}{n!}\,,\quad
\frac{P_n^{(\al,\be)}(-1)}{P_n^{(\al,\be)}(1)}=\frac{(-1)^n(\be+1)_n}{(\al+1)_n}\,.
\label{50}
\end{equation}
Use (9.8.1) and \eqref{48} or see \mycite{DLMF}{Table 18.6.1}.
%
\paragraph{Generating functions}
Formula (9.8.15) was first obtained by Brafman \myciteKLS{109}.
%
\paragraph{Bilateral generating functions}
For $0\le r<1$ and $x,y\in[-1,1]$ we have in terms of $F_4$ (see~\eqref{62}):
\begin{align}
&\sum_{n=0}^\iy\frac{(\al+\be+1)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\,
P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y)
=\frac1{(1+r)^{\al+\be+1}}
\nonumber\\
&\qquad\quad\times F_4\Big(\thalf(\al+\be+1),\thalf(\al+\be+2);\al+1,\be+1;
\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big),
\label{58}\sLP
&\sum_{n=0}^\iy\frac{2n+\al+\be+1}{n+\al+\be+1}
\frac{(\al+\be+2)_n\,n!}{(\al+1)_n(\be+1)_n}\,r^n\,
P_n^{(\al,\be)}(x)\,P_n^{(\al,\be)}(y)
=\frac{1-r}{(1+r)^{\al+\be+2}}\nonumber\\
&\qquad\quad\times F_4\Big(\thalf(\al+\be+2),\thalf(\al+\be+3);\al+1,\be+1;
\frac{r(1-x)(1-y)}{(1+r)^2},\frac{r(1+x)(1+y)}{(1+r)^2}\Big).
\label{59}
\end{align}
Formulas \eqref{58} and \eqref{59} were first
given by Bailey \myciteKLS{91}{(2.1), (2.3)}.
See Stanton \myciteKLS{485} for a shorter proof.
(However, in the second line of
\myciteKLS{485}{(1)} $z$ and $Z$ should be interchanged.)$\;$
As observed in Bailey \myciteKLS{91}{p.10}, \eqref{59} follows
from \eqref{58}
by applying the operator $r^{-\half(\al+\be-1)}\,\frac d{dr}\circ r^{\half(\al+\be+1)}$
to both sides of \eqref{58}.
In view of \eqref{60}, formula \eqref{59} is the Poisson kernel for Jacobi
polynomials. The \RHS\ of \eqref{59} makes clear that this kernel is positive.
See also the discussion in Askey \myciteKLS{46}{following (2.32)}.
%
\paragraph{Quadratic transformations}
\begin{align}
\frac{C_{2n}^{(\al+\half)}(x)}{C_{2n}^{(\al+\half)}(1)}
=\frac{P_{2n}^{(\al,\al)}(x)}{P_{2n}^{(\al,\al)}(1)}
&=\frac{P_n^{(\al,-\half)}(2x^2-1)}{P_n^{(\al,-\half)}(1)}\,,
\label{51}\\
\frac{C_{2n+1}^{(\al+\half)}(x)}{C_{2n+1}^{(\al+\half)}(1)}
=\frac{P_{2n+1}^{(\al,\al)}(x)}{P_{2n+1}^{(\al,\al)}(1)}
&=\frac{x\,P_n^{(\al,\half)}(2x^2-1)}{P_n^{(\al,\half)}(1)}\,.
\label{52}
\end{align}
See p.221, Remarks, last two formulas together with \eqref{50} and \eqref{49}.
Or see \mycite{DLMF}{(18.7.13), (18.7.14)}.
%
\paragraph{Differentiation formulas}
Each differentiation formula is given in two equivalent forms.
\begin{equation}
\begin{split}
\frac d{dx}\left((1-x)^\al P_n^{(\al,\be)}(x)\right)&=
-(n+\al)\,(1-x)^{\al-1} P_n^{(\al-1,\be+1)}(x),\\
\left((1-x)\frac d{dx}-\al\right)P_n^{(\al,\be)}(x)&=
-(n+\al)\,P_n^{(\al-1,\be+1)}(x).
\end{split}
\label{68}
\end{equation}
%
\begin{equation}
\begin{split}
\frac d{dx}\left((1+x)^\be P_n^{(\al,\be)}(x)\right)&=
(n+\be)\,(1+x)^{\be-1} P_n^{(\al+1,\be-1)}(x),\\
\left((1+x)\frac d{dx}+\be\right)P_n^{(\al,\be)}(x)&=
(n+\be)\,P_n^{(\al+1,\be-1)}(x).
\end{split}
\label{69}
\end{equation}
Formulas \eqref{68} and \eqref{69} follow from
\mycite{DLMF}{(15.5.4), (15.5.6)}
together with (9.8.1). They also follow from each other by \eqref{48}.
%
\paragraph{Generalized Gegenbauer polynomials}
These are defined by
\begin{equation}
S_{2m}^{(\al,\be)}(x):=\const P_m^{(\al,\be)}(2x^2-1),\qquad
S_{2m+1}^{(\al,\be)}(x):=\const x\,P_m^{(\al,\be+1)}(2x^2-1)
\label{70}
\end{equation}
in the notation of \myciteKLS{146}{p.156}
(see also \cite{K27}), while \cite[Section 1.5.2]{K26}
has $C_n^{(\la,\mu)}(x)=\const\allowbreak\times S_n^{(\la-\half,\mu-\half)}(x)$.
For $\al,\be>-1$ we have the orthogonality relation
\begin{equation}
\int_{-1}^1 S_m^{(\al,\be)}(x)\,S_n^{(\al,\be)}(x)\,|x|^{2\be+1}(1-x^2)^\al\,dx
=0\qquad(m\ne n).
\label{71}
\end{equation}
For $\be=\al-1$ generalized Gegenbauer polynomials are limit cases of
continuous $q$-ultraspherical polynomials, see \eqref{176}.
If we define the {\em Dunkl operator} $T_\mu$ by
\begin{equation}
(T_\mu f)(x):=f'(x)+\mu\,\frac{f(x)-f(-x)}x
\label{72}
\end{equation}
and if we choose the constants in \eqref{70} as
\begin{equation}
S_{2m}^{(\al,\be)}(x)=\frac{(\al+\be+1)_m}{(\be+1)_m}\, P_m^{(\al,\be)}(2x^2-1),\quad
S_{2m+1}^{(\al,\be)}(x)=\frac{(\al+\be+1)_{m+1}}{(\be+1)_{m+1}}\,
x\,P_m^{(\al,\be+1)}(2x^2-1)
\label{73}
\end{equation}
then (see \cite[(1.6)]{K5})
\begin{equation}
T_{\be+\half}S_n^{(\al,\be)}=2(\al+\be+1)\,S_{n-1}^{(\al+1,\be)}.
\label{74}
\end{equation}
Formula \eqref{74} with \eqref{73} substituted gives rise to two
differentiation formulas involving Jacobi polynomials which are equivalent to
(9.8.7) and \eqref{69}.
Composition of \eqref{74} with itself gives
\[
T_{\be+\half}^2S_n^{(\al,\be)}=4(\al+\be+1)(\al+\be+2)\,S_{n-2}^{(\al+2,\be)},
\]
which is equivalent to the composition of (9.8.7) and \eqref{69}:
\begin{equation}
\left(\frac{d^2}{dx^2}+\frac{2\be+1}x\,\frac d{dx}\right)P_n^{(\al,\be)}(2x^2-1)
=4(n+\al+\be+1)(n+\be)\,P_{n-1}^{(\al+2,\be)}(2x^2-1).
\label{75}
\end{equation}
Formula \eqref{75} was also given in \myciteKLS{322}{(2.4)}.
%
\subsection*{9.8.1 Gegenbauer / Ultraspherical}
\label{sec9.8.1}
%
\paragraph{Notation}
Here the Gegenbauer polynomial is denoted by $C_n^\la$ instead of $C_n^{(\la)}$.
%
\paragraph{Orthogonality relation}
Write the \RHS\ of (9.8.20) as $h_n\,\de_{m,n}$. Then
\begin{equation}
\frac{h_n}{h_0}=
\frac\la{\la+n}\,\frac{(2\la)_n}{n!}\,,\quad
h_0=\frac{\pi^\half\,\Ga(\la+\thalf)}{\Ga(\la+1)},\quad
\frac{h_n}{h_0\,(C_n^\la(1))^2}=
\frac\la{\la+n}\,\frac{n!}{(2\la)_n}\,.
\label{61}
\end{equation}
%
\paragraph{Hypergeometric representation}
Beside (9.8.19) we have also
\begin{equation}
C_n^\lambda(x)=\sum_{\ell=0}^{\lfloor n/2\rfloor}\frac{(-1)^{\ell}(\lambda)_{n-\ell}}
{\ell!\;(n-2\ell)!}\,(2x)^{n-2\ell}
=(2x)^{n}\,\frac{(\lambda)_{n}}{n!}\,
\hyp21{-\thalf n,-\thalf n+\thalf}{1-\la-n}{\frac1{x^2}}.
\label{57}
\end{equation}
See \mycite{DLMF}{(18.5.10)}.
%
\paragraph{Special value}
\begin{equation}
C_n^{\la}(1)=\frac{(2\la)_n}{n!}\,.
\label{49}
\end{equation}
Use (9.8.19) or see \mycite{DLMF}{Table 18.6.1}.
%
\paragraph{Expression in terms of Jacobi}
%
\begin{equation}
\frac{C_n^\la(x)}{C_n^\la(1)}=
\frac{P_n^{(\la-\half,\la-\half)}(x)}{P_n^{(\la-\half,\la-\half)}(1)}\,,\qquad
C_n^\la(x)=\frac{(2\la)_n}{(\la+\thalf)_n}\,P_n^{(\la-\half,\la-\half)}(x).
\label{65}
\end{equation}
%
\paragraph{Re: (9.8.21)}
By iteration of recurrence relation (9.8.21):
\begin{multline}
x^2 C_n^\la(x)=
\frac{(n+1)(n+2)}{4(n+\la)(n+\la+1)}\,C_{n+2}^\la(x)+
\frac{n^2+2n\la+\la-1}{2(n+\la-1)(n+\la+1)}\,C_n^\la(x)\\
+\frac{(n+2\la-1)(n+2\la-2)}{4(n+\la)(n+\la-1)}\,C_{n-2}^\la(x).
\label{6}
\end{multline}
%
\paragraph{Bilateral generating functions}
\begin{multline}
\sum_{n=0}^\iy\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y)
=\frac1{(1-2rxy+r^2)^\la}\,\hyp21{\thalf\la,\thalf(\la+1)}{\la+\thalf}
{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\\
(r\in(-1,1),\;x,y\in[-1,1]).
\label{66}
\end{multline}
For the proof put $\be:=\al$ in \eqref{58}, then use \eqref{63} and \eqref{65}.
The Poisson kernel for Gegenbauer polynomials can be derived in a similar way
from \eqref{59}, or alternatively by applying the operator
$r^{-\la+1}\frac d{dr}\circ r^\la$ to both sides of \eqref{66}:
\begin{multline}
\sum_{n=0}^\iy\frac{\la+n}\la\,\frac{n!}{(2\la)_n}\,r^n\,C_n^\la(x)\,C_n^\la(y)
=\frac{1-r^2}{(1-2rxy+r^2)^{\la+1}}\\
\times\hyp21{\thalf(\la+1),\thalf(\la+2)}{\la+\thalf}
{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}}\qquad
(r\in(-1,1),\;x,y\in[-1,1]).
\label{67}
\end{multline}
Formula \eqref{67} was obtained by Gasper \& Rahman \myciteKLS{234}{(4.4)}
as a limit case of their formula for the Poisson kernel for continuous
$q$-ultraspherical polynomials.
%
\paragraph{Trigonometric expansions}
By \mycite{DLMF}{(18.5.11), (15.8.1)}:
\begin{align}
C_n^{\la}(\cos\tha)
&=\sum_{k=0}^n\frac{(\la)_k(\la)_{n-k}}{k!\,(n-k)!}\,e^{i(n-2k)\tha}
=e^{in\tha}\frac{(\la)_n}{n!}\,
\hyp21{-n,\la}{1-\la-n}{e^{-2i\tha}}\label{103}\\
&=\frac{(\la)_n}{2^\la n!}\,
e^{-\half i\la\pi}e^{i(n+\la)\tha}\,(\sin\tha)^{-\la}\,
\hyp21{\la,1-\la}{1-\la-n}{\frac{i e^{-i\tha}}{2\sin\tha}}\label{104}\\
&=\frac{(\la)_n}{n!}\,\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1-\la-n)_k k!}\,
\frac{\cos((n-k+\la)\tha+\thalf(k-\la)\pi)}{(2\sin\tha)^{k+\la}}\,.\label{105}
\end{align}
In \eqref{104} and \eqref{105} we require that
$\tfrac16\pi<\tha<\tfrac56\pi$. Then the convergence is absolute for $\la>\thalf$
and conditional for $0<\la\le\thalf$.
By \mycite{DLMF}{(14.13.1), (14.3.21), (15.8.1)]}:
\begin{align}
C_n^\la(\cos\tha)&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\,
\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\,
\sum_{k=0}^\iy\frac{(1-\la)_k(n+1)_k}{(n+\la+1)_k k!}\,
\sin\big((2k+n+1)\tha\big)
\label{7}\\
&=\frac{2\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\,
\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{1-2\la}\,
\Im\!\!\left(e^{i(n+1)\tha}\,\hyp21{1-\la,n+1}{n+\la+1}{e^{2i\tha}}\right)\nonumber\\
&=\frac{2^\la\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\,
\frac{(2\la)_n}{(\la+1)_n}\,(\sin\tha)^{-\la}\,
\Re\!\!\left(e^{-\thalf i\la\pi}e^{i(n+\la)\tha}\,
\hyp21{\la,1-\la}{1+\la+n}{\frac{e^{i\tha}}{2i\sin\tha}}\right)\nonumber\\
&=\frac{2^{2\la}\Ga(\la+\thalf)}{\pi^\half\Ga(\la+1)}\,\frac{(2\la)_n}{(\la+1)_n}\,
\sum_{k=0}^\iy\frac{(\la)_k(1-\la)_k}{(1+\la+n)_k k!}\,
\frac{\cos((n+k+\la)\tha-\thalf(k+\la)\pi)}{(2\sin\tha)^{k+\la}}\,.
\label{106}
\end{align}
We require that $0<\tha<\pi$ in \eqref{7} and $\tfrac16\pi<\tha<\tfrac56\pi$ in
\eqref{106} The convergence is absolute for $\la>\thalf$ and conditional for
$0<\la\le\thalf$.
For $\la\in\Zpos$ the above series terminate after the term with
$k=\la-1$.
Formulas \eqref{7} and \eqref{106} are also given in
\mycite{Sz}{(4.9.22), (4.9.25)}.
%
\paragraph{Fourier transform}
\begin{equation}
\frac{\Ga(\la+1)}{\Ga(\la+\thalf)\,\Ga(\thalf)}\,
\int_{-1}^1 \frac{C_n^\la(y)}{C_n^\la(1)}\,(1-y^2)^{\la-\half}\,
e^{ixy}\,dy
=i^n\,2^\la\,\Ga(\la+1)\,x^{-\la}\,J_{\la+n}(x).
\label{8}
\end{equation}
See \mycite{DLMF}{(18.17.17) and (18.17.18)}.
%
\paragraph{Laplace transforms}
\begin{equation}
\frac2{n!\,\Ga(\la)}\,
\int_0^\iy H_n(tx)\,t^{n+2\la-1}\,e^{-t^2}\,dt=C_n^\la(x).
\label{56}
\end{equation}
See Nielsen \cite[p.48, (4) with p.47, (1) and p.28, (10)]{K4} (1918)
or Feldheim \cite[(28)]{K3} (1942).
\begin{equation}
\frac2{\Ga(\la+\thalf)}\,\int_0^1 \frac{C_n^\la(t)}{C_n^\la(1)}\,
(1-t^2)^{\la-\half}\,t^{-1}\,(x/t)^{n+2\la+1}\,e^{-x^2/t^2}\,dt
=2^{-n}\,H_n(x)\,e^{-x^2}\quad(\la>-\thalf).
\label{46}
\end{equation}
Use Askey \& Fitch \cite[(3.29)]{K2} for $\al=\pm\thalf$ together with
\eqref{48}, \eqref{51}, \eqref{52}, \eqref{54} and \eqref{55}.
\paragraph{Addition formula} (see \mycite{AAR}{(9.8.5$'$)]})
\begin{multline}
R_n^{(\al,\al)}\big(xy+(1-x^2)^\half(1-y^2)^\half t\big)
=\sum_{k=0}^n \frac{(-1)^k(-n)_k\,(n+2\al+1)_k}{2^{2k}((\al+1)_k)^2}\\
\times(1-x^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(x)\,(1-y^2)^{k/2} R_{n-k}^{(\al+k,\al+k)}(y)\,
\om_k^{(\al-\half,\al-\half)}\,R_k^{(\al-\half,\al-\half)}(t),
\label{108}
\end{multline}
where
\[
R_n^{(\al,\be)}(x):=P_n^{(\al,\be)}(x)/P_n^{(\al,\be)}(1),\quad
\om_n^{(\al,\be)}:=\frac{\int_{-1}^1 (1-x)^\al(1+x)^\be\,dx}
{\int_{-1}^1 (R_n^{(\al,\be)}(x))^2\,(1-x)^\al(1+x)^\be\,dx}\,.
\]
%
\subsection*{9.8.2 Chebyshev}
\label{sec9.8.2}
In addition to the Chebyshev polynomials $T_n$ of the first kind (9.8.35)
and $U_n$ of the second kind (9.8.36),
\begin{align}
T_n(x)&:=\frac{P_n^{(-\half,-\half)}(x)}{P_n^{(-\half,-\half)}(1)}
=\cos(n\tha),\quad x=\cos\tha,\\
U_n(x)&:=(n+1)\,\frac{P_n^{(\half,\half)}(x)}{P_n^{(\half,\half)}(1)}
=\frac{\sin((n+1)\tha)}{\sin\tha}\,,\quad x=\cos\tha,
\end{align}
we have Chebyshev polynomials $V_n$ {\em of the third kind}
and $W_n$ {\em of the fourth kind},
\begin{align}
V_n(x)&:=\frac{P_n^{(-\half,\half)}(x)}{P_n^{(-\half,\half)}(1)}
=\frac{\cos((n+\thalf)\tha)}{\cos(\thalf\tha)}\,,\quad x=\cos\tha,\\
W_n(x)&:=(2n+1)\,\frac{P_n^{(\half,-\half)}(x)}{P_n^{(\half,-\half)}(1)}
=\frac{\sin((n+\thalf)\tha)}{\sin(\thalf\tha)}\,,\quad x=\cos\tha,
\end{align}
see \cite[Section 1.2.3]{K20}. Then there is the symmetry
\begin{equation}
V_n(-x)=(-1)^n W_n(x).
\label{140}
\end{equation}
The names of Chebyshev polynomials of the third and fourth kind
and the notation $V_n(x)$ are due to Gautschi \cite{K21}.
The notation $W_n(x)$ was first used by Mason \cite{K22}.
Names and notations for Chebyshev polynomials of the third and fourth
kind are interchanged in \mycite{AAR}{Remark 2.5.3} and
\mycite{DLMF}{Table 18.3.1}.
%
\subsection*{9.9 Pseudo Jacobi (or Routh-Romanovski)}
\label{sec9.9}
In this section in \mycite{KLS} the pseudo Jacobi polynomial $P_n(x;\nu,N)$ in (9.9.1)
is considered
for $N\in\ZZ_{\ge0}$ and $n=0,1,\ldots,n$. However, we can more generally take
$-\thalf<N\in\RR$ (so here I overrule my convention formulated in the
beginning of this paper), $N_0$ integer such that $N-\thalf\le N_0<N+\thalf$, and $n=0,1,\ldots,N_0$
(see \myciteKLS{382}{\S5, case A.4}). The orthogonality relation (9.9.2)
is valid for $m,n=0,1,\ldots,N_0$.
%
\paragraph{History}
These polynomials were first obtained by Routh \cite{K13} in 1885, and later, independently,
by Romanovski \myciteKLS{463} in 1929.
%
\paragraph{Limit relation:}
{\bf Pseudo big $q$-Jacobi $\longrightarrow$ Pseudo Jacobi}\\
See also \eqref{118}.
%
\paragraph{References}
See also \mycite{Ism}{\S20.1}, \myciteKLS{51},
\myciteKLS{384}, \cite{K11}, \cite{K10}, \cite{K12}.
%
\subsection*{9.10 Meixner}
\label{sec9.10}
\paragraph{History}
In 1934 Meixner \myciteKLS{406} (see
(1.1) and case IV on pp.~10, 11 and 12) gave the orthogonality
measure for the polynomials $P_n$ given by the generating function
\[
e^{x u(t)}\,f(t)=\sum_{n=0}^\iy P_n(x)\,\frac{t^n}{n!}\,,
\]
where
\[
e^{u(t)}=\left(\frac{1-\be t}{1-\al t}\right)^{\frac1{\al-\be}},\quad
f(t)=\frac{(1-\be t)^{\frac{k_2}{\be(\al-\be)}}}{(1-\al t)^{\frac{k_2}{\al(\al-\be)}}}\quad
(k_2<0;\;\al>\be>0\;\;{\rm or}\;\;\al<\be<0).
\]
Then $P_n$ can be expressed as a Meixner polynomial:
\[
P_n(x)=(-k_2(\al\be)^{-1})_n\,\be^n\,
M_n\left(-\,\frac{x+k_2\al^{-1}}{\al-\be},-k_2(\al\be)^{-1},\be\al^{-1}\right).
\]
In 1938 Gottlieb \cite[\S2]{K1} introduces polynomials $l_n$ ``of Laguerre type''
which turn out to be special Meixner polynomials:
$l_n(x)=e^{-n\la} M_n(x;1,e^{-\la})$.
%
\paragraph{Uniqueness of orthogonality measure}
The coefficient of $p_{n-1}(x)$ in (9.10.4) behaves as $O(n^2)$ as $n\to\iy$.
Hence \eqref{93} holds, by which the orthogonality measure is unique.
%
\subsection*{9.11 Krawtchouk}
\label{sec9.11}
%
\paragraph{Special values}
By (9.11.1) and the binomial formula:
\begin{equation}
K_n(0;p,N)=1,\qquad
K_n(N;p,N)=(1-p^{-1})^n.