Pollaczek polynomials

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11: 18.26 Wilson Class: Continued

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18.26.8 $$\underset{t\to \mathrm{\infty }}{lim}{S}_n({(x-t)}^2;\lambda +it,\lambda -it,t\cot \varphi )/t^n=n!{(\csc \varphi )}^n{P}_n^{(\lambda )}(x;\varphi ).$$

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Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $S_n(x;a,b,c)$: continuous dual Hahn polynomial, $\csc z$: cosecant function, $\cot z$: cotangent function, $!$: factorial (as in $n!$), $t$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

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12: 15.9 Relations to Other Functions

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Meixner–Pollaczek

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15.9.10 $$P_n^{(\lambda )}(x;\varphi )=\frac{{\left(2\lambda \right)}_n}{n!}{\mathrm{e}}^{n\mathrm{i}\varphi }F(\genfrac{}{}{0pt}{}{-n,\lambda +\mathrm{i}x}{2\lambda };1-{\mathrm{e}}^{-2\mathrm{i}\varphi }).$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $x$: real variable and $n$: integer

Permalink:

http://dlmf.nist.gov/15.9.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

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13: Errata

… ►This release is the result of that decision and it includes, among other new material, enlarged sections on associated OP’s, Pollaczek polynomials and physical applications. … ►

18.35.1

$P_{-1}^{(\lambda )}(x;a,b,c)=0,$

$P_0^{(\lambda )}(x;a,b,c)=1$

These equations which were previously given for Pollaczek polynomials of type 2 has been updated for Pollaczek polynomials of type 3.

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18.35.2 $$P_{n+1}^{(\lambda )}(x;a,b,c)=\frac{2(n+c+\lambda +a)x+2b}{n+c+1}{P}_n^{(\lambda )}(x;a,b,c)-\frac{n+c+2\lambda -1}{n+c+1}{P}_{n-1}^{(\lambda )}(x;a,b,c),$$ $n=0,1,\mathrm{\dots }$

This recurrence relation which was previously given for Pollaczek polynomials of type 2 (the case $c=0$) has been updated for Pollaczek polynomials of type 3.

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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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14: DLMF Project News

error generating summary

15: Bibliography K

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Meixner-Pollaczek polynomials and the Heisenberg algebra. J. Math. Phys. 30 (4), pp. 767–769.

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External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§18.21(ii).

Encodings:

BibTeX

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16: Bibliography L

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On the asymptotics of the Meixner-Pollaczek polynomials and their zeros. Constr. Approx. 17 (1), pp. 59–90.

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External Links:

ISSN 0176-4276, Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§18.24.

Encodings:

BibTeX

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17: 18.2 General Orthogonal Polynomials

… ►This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials (§18.20(i)). … ►The generating functions (18.12.13), (18.12.15), (18.23.3), (18.23.4), (18.23.5) and (18.23.7) for Laguerre, Hermite, Krawtchouk, Meixner, Charlier and Meixner–Pollaczek polynomials, respectively, can be written in the form (18.2.45). …

18: Bibliography B

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Asymptotic behavior of the Pollaczek polynomials and their zeros. Stud. Appl. Math. 96, pp. 307–338.

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External Links:

ISSN 0022-2526, MathReview (Harold Exton), ZentralBlatt

Referenced by:

§18.35(iii).

Encodings:

BibTeX

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19: 18.40 Methods of Computation

… ►The example chosen is inversion from the ${\alpha }_n,{\beta }_n$ for the weight function for the repulsive Coulomb–Pollaczek, RCP, polynomials of (18.39.50). …

20: Bibliography P

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Zonal Polynomials of Order $1$ Through $12$ . In Selected Tables in Mathematical Statistics, H. L. Harter and D. B. Owen (Eds.), Vol. 2, pp. 199–388.

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External Links:

MathReview

Referenced by:

§35.11.

Encodings:

BibTeX

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Orthogonal polynomials and some $q$-beta integrals of Ramanujan. J. Math. Anal. Appl. 112 (2), pp. 517–540.

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External Links:

ISSN 0022-247X, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.33(iv), §18.33(v).

Encodings:

BibTeX

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Sur une généralisation des polynomes de Legendre. C. R. Acad. Sci. Paris 228, pp. 1363–1365.

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External Links:

ISSN 0001-4036, MathReview (G. Szegö)

Referenced by:

§18.35(i).

Encodings:

BibTeX

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Systèmes de polynomes biorthogonaux qui généralisent les polynomes ultrasphériques. C. R. Acad. Sci. Paris 228, pp. 1998–2000.

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External Links:

ISSN 0001-4036, MathReview (G. Szegö)

Referenced by:

§18.35(i).

Encodings:

BibTeX

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Sur une famille de polynômes orthogonaux à quatre paramètres. C. R. Acad. Sci. Paris 230, pp. 2254–2256.

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External Links:

MathReview (A. Erdélyi)

Referenced by:

(18.30.17), (18.30.18), (18.35.6_5), (18.35.6_6), §18.35(i).

Encodings:

BibTeX

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