§18.35 Pollaczek Polynomials

Keywords:: Pollaczek polynomials
Referenced by:: ¶ ‣ §18.30
Permalink:: http://dlmf.nist.gov/18.35

§18.35(i) Definition and Hypergeometric Representation
§18.35(ii) Orthogonality
§18.35(iii) Other Properties

§18.35(i) Definition and Hypergeometric Representation

Notes:: See Ismail (2005, (5.4.1), (5.4.9), (5.4.10)).
Keywords:: Pollaczek polynomials, hypergeometric function
Permalink:: http://dlmf.nist.gov/18.35.i

18.35.1

$\mathop{P^{{(\lambda)}}_{{-1}}\/}\nolimits\!\left(x;a,b\right)=0,$

$\mathop{P^{{(\lambda)}}_{{0}}\/}\nolimits\!\left(x;a,b\right)=1,$

Symbols:: $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;a,b\right)$ : Pollaczek polynomial and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.35.E1
Encodings:: TeX, TeX, pMML, pMML, png, png

and

18.35.2 $(n+1)\mathop{P^{{(\lambda)}}_{{n+1}}\/}\nolimits\!\left(x;a,b\right)={2((n+\lambda+a)x+b)}\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;a,b\right)-{(n+2\lambda-1)}\mathop{P^{{(\lambda)}}_{{n-1}}\/}\nolimits\!\left(x;a,b\right),$ $n=0,1,\dots$ .

Symbols:: $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;a,b\right)$ : Pollaczek polynomial, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.35.E2
Encodings:: TeX, pMML, png

Next, let

18.35.3 $\tau _{{a,b}}(\theta)=\frac{a\mathop{\cos\/}\nolimits\theta+b}{\mathop{\sin\/}\nolimits\theta},$ $0<\theta<\pi$ .

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and $\tau _{{a,b}}(\theta)$
Permalink:: http://dlmf.nist.gov/18.35.E3
Encodings:: TeX, pMML, png

Then

18.35.4 $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta;a,b\right)=\frac{\left(\lambda-i\tau _{{a,b}}(\theta)\right)_{{n}}}{n!}e^{{in\theta}}\*\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({-n,\lambda+i\tau _{{a,b}}(\theta)\atop-n-\lambda+1+i\tau _{{a,b}}(\theta)};e^{{-2i\theta}}\right)=\sum _{{\ell=0}}^{n}\frac{\left(\lambda+i\tau _{{a,b}}(\theta)\right)_{{\ell}}}{\ell!}\frac{\left(\lambda-i\tau _{{a,b}}(\theta)\right)_{{n-\ell}}}{(n-\ell)!}e^{{i(n-2\ell)\theta}}.$

Defines:: $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;a,b\right)$ : Pollaczek polynomial
Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\ell$ : 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

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

§18.35(ii) Orthogonality

Notes:: See Ismail (2005, (5.4.12), (5.4.13)).
Keywords:: Pollaczek polynomials
Permalink:: http://dlmf.nist.gov/18.35.ii

18.35.5 ${\int _{{-1}}^{1}\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;a,b\right)\mathop{P^{{(\lambda)}}_{{m}}\/}\nolimits\!\left(x;a,b\right)w^{{(\lambda)}}(x;a,b)dx=0},$ $n\neq m$ ,

Symbols:: $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;a,b\right)$ : 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

where

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $\mathop{\sin\/}\nolimits z$ : sine function, $w(x)$ : weight function and $\tau _{{a,b}}(\theta)$
Permalink:: http://dlmf.nist.gov/18.35.E6
Encodings:: TeX, pMML, png

§18.35(iii) Other Properties

Notes:: See Ismail (2005, (5.4.3), (5.4.8)).
Keywords:: Pollaczek polynomials
Permalink:: http://dlmf.nist.gov/18.35.iii

18.35.7 $(1-ze^{{i\theta}})^{{-\lambda+i\tau _{{a,b}}(\theta)}}(1-ze^{{-i\theta}})^{{-\lambda-i\tau _{{a,b}}(\theta)}}=\sum _{{n=0}}^{\infty}\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta;a,b\right)z^{n},$ $|z|<1$ , $0<\theta<\pi$ .

Symbols:: $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;a,b\right)$ : Pollaczek polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $z$ : complex variable, $n$ : nonnegative integer and $\tau _{{a,b}}(\theta)$
Permalink:: http://dlmf.nist.gov/18.35.E7
Encodings:: TeX, pMML, png

18.35.8 $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;0,0\right)=\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;a,b\right)$ : Pollaczek polynomial, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.35.E8
Encodings:: TeX, pMML, png

18.35.9 $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\phi;0,x\mathop{\sin\/}\nolimits\phi\right)=\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right).$

Symbols:: $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right)$ : Meixner–Pollaczek polynomial, $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;a,b\right)$ : Pollaczek polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.35.E9
Encodings:: TeX, pMML, png

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

See Bo and Wong (1996) for an asymptotic expansion of $\mathop{P^{{(\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits(n^{{-\frac{1}{2}}}\theta);a,b\right)$ as $n\to\infty$ , with $a$ and $b$ fixed. This expansion is in terms of the Airy function $\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ 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 $\mathop{P^{{(\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits(n^{{-\frac{1}{2}}}\theta);a,b\right)$ .

§18.35 Pollaczek Polynomials

Contents

§18.35(i) Definition and Hypergeometric Representation

§18.35(ii) Orthogonality

§18.35(iii) Other Properties