§18.35 Pollaczek Polynomials

18.35.1		$P_{-1}^{(\lambda )}(x;a,b)$	$=0,$
		$P_0^{(\lambda )}(x;a,b)$	$=1,$
ⓘ Symbols: $P_n^{(\lambda )}(x;a,b)$: 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		$$(n+1)P_{n+1}^{(\lambda )}(x;a,b)=2((n+\lambda +a)x+b)P_n^{(\lambda )}(x;a,b)-(n+2\lambda -1)P_{n-1}^{(\lambda )}(x;a,b),$$
$n=0,1,\mathrm{\dots }$.
ⓘ Symbols: $P_n^{(\lambda )}(x;a,b)$: 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		$${\tau }_{a,b}(\theta )=\frac{a\cos \theta +b}{\sin \theta },$$
.
ⓘ Defines: ${\tau }_{a,b}(\theta )$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $\sin z$: 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		$$\begin{array}{ll}{P}_n^{(\lambda )}(\cos \theta ;a,b)& =\frac{{\left(\lambda -\mathrm{i}{\tau }_{a,b}(\theta )\right)}_n}{n!}{\mathrm{e}}^{\mathrm{i}n\theta }{}_{2}F_1(\genfrac{}{}{0pt}{}{-n,\lambda +\mathrm{i}{\tau }_{a,b}(\theta )}{-n-\lambda +1+\mathrm{i}{\tau }_{a,b}(\theta )};{\mathrm{e}}^{-2\mathrm{i}\theta })\\ & =\sum _{\mathrm{\ell }=0}^n\frac{{\left(\lambda +\mathrm{i}{\tau }_{a,b}(\theta )\right)}_{\mathrm{\ell }}}{\mathrm{\ell }!}\frac{{\left(\lambda -\mathrm{i}{\tau }_{a,b}(\theta )\right)}_{n-\mathrm{\ell }}}{(n-\mathrm{\ell })!}{\mathrm{e}}^{\mathrm{i}(n-2\mathrm{\ell })\theta }.\end{array}$$
ⓘ Defines: $P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial Symbols: ${}_{2}F_1(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: 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).

18.35.5		$${\int }_{-1}^1{P}_n^{(\lambda )}(x;a,b)P_m^{(\lambda )}(x;a,b)w^{(\lambda )}(x;a,b)dx=0,$$
$n\ne m$,
ⓘ Symbols: $P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $dx$: differential of $x$, $\int$: integral, $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.35.E5 Encodings: TeX, pMML, png See also: Annotations for §18.35(ii), §18.35 and Ch.18

where

18.35.6		$$w^{(\lambda )}(\cos \theta ;a,b)=\begin{array}{l}{\pi }^{-1}{2}^{2\lambda -1}{\mathrm{e}}^{(2\theta -\pi ){\tau }_{a,b}(\theta )}\\ \times {(\sin \theta )}^{2\lambda -1}{\left|\mathrm{\Gamma }\left(\lambda +\mathrm{i}{\tau }_{a,b}(\theta )\right)\right|}^2,\end{array}$$
$a\ge b\ge -a$, $\lambda >-\frac{1}{2}$, .
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $w(x)$: weight function and ${\tau }_{a,b}(\theta )$ Permalink: http://dlmf.nist.gov/18.35.E6 Encodings: TeX, pMML, png See also: Annotations for §18.35(ii), §18.35 and Ch.18

18.35.7		$${(1-z{\mathrm{e}}^{\mathrm{i}\theta })}^{-\lambda +\mathrm{i}{\tau }_{a,b}(\theta )}{(1-z{\mathrm{e}}^{-\mathrm{i}\theta })}^{-\lambda -\mathrm{i}{\tau }_{a,b}(\theta )}=\sum _{n=0}^{\mathrm{\infty }}{P}_n^{(\lambda )}(\cos \theta ;a,b)z^n,$$
, .
ⓘ Symbols: $P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $n$: nonnegative integer and ${\tau }_{a,b}(\theta )$ Permalink: http://dlmf.nist.gov/18.35.E7 Encodings: TeX, pMML, png See also: Annotations for §18.35(iii), §18.35 and Ch.18

18.35.8		$$P_n^{(\lambda )}(x;0,0)=C_n^{(\lambda )}\left(x\right),$$
ⓘ Symbols: $P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.35.E8 Encodings: TeX, pMML, png See also: Annotations for §18.35(iii), §18.35 and Ch.18

18.35.9		$$P_n^{(\lambda )}(\cos \varphi ;0,x\sin \varphi )=P_n^{(\lambda )}(x;\varphi ).$$
ⓘ Symbols: $P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $\cos z$: cosine function, $\sin z$: sine function, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.35.E9 Encodings: TeX, pMML, png See also: Annotations for §18.35(iii), §18.35 and Ch.18

For the polynomials $C_n^{(\lambda )}\left(x\right)$ and $P_n^{(\lambda )}(x;\varphi )$ see §§18.3 and 18.19, respectively.

See Bo and Wong (1996) for an asymptotic expansion of $P_n^{(\frac{1}{2})}(\cos \left(n^{-\frac{1}{2}}\theta \right);a,b)$ as $n\to \mathrm{\infty }$, 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 $(0,\mathrm{\infty })$. Also included is an asymptotic approximation for the zeros of $P_n^{(\frac{1}{2})}(\cos \left(n^{-\frac{1}{2}}\theta \right);a,b)$.