Krawtchouk and q-Racah polynomials

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11: 15.9 Relations to Other Functions

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Krawtchouk

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15.9.8 $$K_n(x;p,N)=F(\genfrac{}{}{0pt}{}{-n,-x}{-N};\frac{1}{p}),$$ $n=0,1,2,\mathrm{\dots },N$;

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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, $K_n(x;p,N)$: Krawtchouk polynomial, $x$: real variable and $n$: integer

Permalink:

http://dlmf.nist.gov/15.9.E8

Encodings:

pMML, png, TeX

See also:

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

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12: 18.2 General Orthogonal Polynomials

… ►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). In fact, these are the only OP’s which are Sheffer polynomials (with Krawtchouk polynomials being only a finite system) …

13: Bibliography L

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A uniform asymptotic expansion for Krawtchouk polynomials. J. Approx. Theory 106 (1), pp. 155–184.

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External Links:

ISSN 0021-9045, Document, MathReview (Ahmed I. Zayed), ZentralBlatt

Referenced by:

§18.24.

Encodings:

BibTeX

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