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1: 18.35 Pollaczek Polynomials

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$P_n^{(\frac{1}{2})}(x;a,b),$

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$P_n^{(\lambda )}(x;a,b)=P_n^{(\lambda )}(x;a,b,0),$

… ►More generally, the $P_n^{(\lambda )}(x;a,b)$ are OP’s if and only if one of the following three conditions holds (in case (iii) work with the monic polynomials (18.35.2_2)). … ►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. …Also included is an asymptotic approximation for the zeros of $P_n^{(\frac{1}{2})}(\cos \left(n^{-\frac{1}{2}}\theta \right);a,b)$. …

2: 18.39 Applications in the Physical Sciences

… ►Thus the $c_N(x)=P_N^{(l+1)}(x;\frac{2Z}{s},-\frac{2Z}{s})$ and the eigenvalues …are determined by the $N$ zeros, $x_i^N$ of the Pollaczek polynomial $P_N^{(l+1)}(x;\frac{2Z}{s},-\frac{2Z}{s})$. … ►The polynomials $P_N^{(l+1)}(x;\frac{2Z}{s},-\frac{2Z}{s})$, for both positive and negative $Z$, define the Coulomb–Pollaczek polynomials (CP OP’s in what follows), see Yamani and Reinhardt (1975, Appendix B, and §IV). … ►

18.39.53 $${\mathrm{\Psi }}_{x_i,l}^N(r)=A_{x_i,l}\sum _{n=0}^{N-1}\frac{n!}{\mathrm{\Gamma }(n+2l+2)}{P}_n^{(l+1)}(x_i;\frac{2Z}{s},-\frac{2Z}{s}){\varphi }_{n,l}(sr),$$ $x_i=\frac{8{ϵ}_i-s^2}{8{ϵ}_i+s^2}$.

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Symbols:

$P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $!$: factorial (as in $n!$), $N$: positive integer, $l$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.39(iv)

Permalink:

http://dlmf.nist.gov/18.39.E53

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(iv), §18.39(iv), §18.39 and Ch.18

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18.39.54 $${\mathrm{\Psi }}_{x,l}(r)=B_l(x)\sum _{n=0}^{\mathrm{\infty }}\frac{n!}{\mathrm{\Gamma }(n+2l+2)}{P}_n^{(l+1)}(x;\frac{2Z}{s},-\frac{2Z}{s}){\varphi }_{n,l}(sr),$$ $x=\frac{8ϵ-s^2}{8ϵ+s^2}$,

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Symbols:

$P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $!$: factorial (as in $n!$), $l$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.39.E54

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(iv), §18.39(iv), §18.39 and Ch.18

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3: 18.1 Notation

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Pollaczek: $P_n^{(\lambda )}(x;a,b)$, $P_n^{(\lambda )}(x;a,b,c)$.

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4: Errata

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18.35.5 $${\int }_{-1}^1{P}_n^{(\lambda )}(x;a,b)P_m^{(\lambda )}(x;a,b)w^{(\lambda )}(x;a,b)dx=\frac{\mathrm{\Gamma }\left(2\lambda +n\right)}{n!(\lambda +a+n)}{\delta }_{n,m},$$ $a\ge b\ge -a$, $\lambda >0$

This equation was updated to give the full normalization. Previously the constraints on $a$, $b$ and $\lambda$ were given in (18.35.6) and included $\lambda >-\frac{1}{2}$. The case is now discussed in (18.35.6_2)–(18.35.6_4).

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18.35.9

$P_n^{(\lambda )}(x;\varphi )=P_n^{(\lambda )}(\cos \varphi ;0,x\sin \varphi ),$

$P_n^{(\lambda )}(\cos \theta ;a,b)=P_n^{(\lambda )}({\tau }_{a,b}(\theta );\theta )$

Previously we gave only the first identity $P_n^{(\lambda )}(\cos \varphi ;0,x\sin \varphi )=P_n^{(\lambda )}(x;\varphi )$.

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