# §18.35 Pollaczek Polynomials

## §18.35(i) Definition and Hypergeometric Representation

 18.35.1 $\displaystyle P^{(\lambda)}_{-1}\left(x;a,b\right)$ $\displaystyle=0,$ $\displaystyle P^{(\lambda)}_{0}\left(x;a,b\right)$ $\displaystyle=1,$ ⓘ Symbols: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\right)$: Pollaczek polynomial and $x$: real variable Permalink: http://dlmf.nist.gov/18.35.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.35(i), §18.35 and Ch.18

and

 18.35.2 $(n+1)P^{(\lambda)}_{n+1}\left(x;a,b\right)={2((n+\lambda+a)x+b)}P^{(\lambda)}_% {n}\left(x;a,b\right)-{(n+2\lambda-1)}P^{(\lambda)}_{n-1}\left(x;a,b\right),$ $n=0,1,\dots$. ⓘ Symbols: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\right)$: Pollaczek polynomial, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.35.E2 Encodings: TeX, pMML, png See also: Annotations for §18.35(i), §18.35 and Ch.18

Next, let

 18.35.3 $\tau_{a,b}(\theta)=\frac{a\cos\theta+b}{\sin\theta},$ $0<\theta<\pi$. ⓘ Defines: $\tau_{a,b}(\theta)$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function and $\sin\NVar{z}$: sine function Permalink: http://dlmf.nist.gov/18.35.E3 Encodings: TeX, pMML, png See also: Annotations for §18.35(i), §18.35 and Ch.18

Then

 18.35.4 $P^{(\lambda)}_{n}\left(\cos\theta;a,b\right)=\frac{{\left(\lambda-\mathrm{i}% \tau_{a,b}(\theta)\right)_{n}}}{n!}e^{\mathrm{i}n\theta}\*{{}_{2}F_{1}}\left({% -n,\lambda+\mathrm{i}\tau_{a,b}(\theta)\atop-n-\lambda+1+\mathrm{i}\tau_{a,b}(% \theta)};e^{-2\mathrm{i}\theta}\right)=\sum_{\ell=0}^{n}\frac{{\left(\lambda+% \mathrm{i}\tau_{a,b}(\theta)\right)_{\ell}}}{\ell!}\frac{{\left(\lambda-% \mathrm{i}\tau_{a,b}(\theta)\right)_{n-\ell}}}{(n-\ell)!}e^{\mathrm{i}(n-2\ell% )\theta}.$ ⓘ Defines: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\right)$: Pollaczek polynomial Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$: $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\cos\NVar{z}$: cosine function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\ell$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $\tau_{a,b}(\theta)$ Permalink: http://dlmf.nist.gov/18.35.E4 Encodings: TeX, pMML, png See also: Annotations for §18.35(i), §18.35 and Ch.18

For the hypergeometric function ${{}_{2}F_{1}}$ see §§15.1, 15.2(i).

## §18.35(ii) Orthogonality

 18.35.5 ${\int_{-1}^{1}P^{(\lambda)}_{n}\left(x;a,b\right)P^{(\lambda)}_{m}\left(x;a,b% \right)w^{(\lambda)}(x;a,b)\mathrm{d}x=0},$ $n\neq m$,

where

 18.35.6 $w^{(\lambda)}(\cos\theta;a,b)=\pi^{-1}\*2^{2\lambda-1}\*e^{(2\theta-\pi)\*\tau% _{a,b}(\theta)}\*(\sin\theta)^{2\lambda-1}\*{\left|\Gamma\left(\lambda+\mathrm% {i}\tau_{a,b}(\theta)\right)\right|^{2}},$ $a\geq b\geq-a$, $\lambda>-\frac{1}{2}$, $0<\theta<\pi$.

## §18.35(iii) Other Properties

 18.35.7 $(1-ze^{\mathrm{i}\theta})^{-\lambda+\mathrm{i}\tau_{a,b}(\theta)}(1-ze^{-% \mathrm{i}\theta})^{-\lambda-\mathrm{i}\tau_{a,b}(\theta)}=\sum_{n=0}^{\infty}% P^{(\lambda)}_{n}\left(\cos\theta;a,b\right)z^{n},$ $|z|<1$, $0<\theta<\pi$.
 18.35.8 $P^{(\lambda)}_{n}\left(x;0,0\right)=C^{(\lambda)}_{n}\left(x\right),$
 18.35.9 $P^{(\lambda)}_{n}\left(\cos\phi;0,x\sin\phi\right)=P^{(\lambda)}_{n}\left(x;% \phi\right).$

For the polynomials $C^{(\lambda)}_{n}\left(x\right)$ and $P^{(\lambda)}_{n}\left(x;\phi\right)$ see §§18.3 and 18.19, respectively.

See Bo and Wong (1996) for an asymptotic expansion of $P^{(\frac{1}{2})}_{n}\left(\cos\left(n^{-\frac{1}{2}}\theta\right);a,b\right)$ as $n\to\infty$, with $a$ and $b$ fixed. This expansion is in terms of the Airy function $\mathrm{Ai}\left(x\right)$ and its derivative (§9.2), and is uniform in any compact $\theta$-interval in $(0,\infty)$. Also included is an asymptotic approximation for the zeros of $P^{(\frac{1}{2})}_{n}\left(\cos\left(n^{-\frac{1}{2}}\theta\right);a,b\right)$.