18 Orthogonal Polynomials Askey Scheme 18.22 Hahn Class: Recurrence Relations and Differences 18.24 Hahn Class: Asymptotic Approximations

§18.23 Hahn Class: Generating Functions

Notes:: For (18.23.3)–(18.23.5), (18.23.7) see Ismail (2005, (6.2.43), (6.1.8), (6.1.22), (5.9.3)). (18.23.1) and (18.23.2) follow by expanding the factors on the left as power series in , and substituting (18.20.5) on the right. (18.23.6) is a limiting case of (18.26.18) via (18.26.6).
Keywords:: Hahn class orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.23

For the definition of generalized hypergeometric functions see §16.2.

¶ Hahn

Keywords:: Hahn class orthogonal polynomials, confluent hypergeometric functions

18.23.1 $\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left({-x\atop\alpha+1};-z\right)% \mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left({x-N\atop\beta+1};z\right)=\sum_{% {n=0}}^{N}\frac{\left(-N\right)_{{n}}}{\left(\beta+1\right)_{{n}}n!}\mathop{Q_% {{n}}\/}\nolimits\!\left(x;\alpha,\beta,N\right)z^{n},$ $x=0,1,\dots,N$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{Q_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,N\right)$ : Hahn polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, : complex variable, : nonnegative integer, : positive integer and : real variable
Referenced by:: §18.23
Permalink:: http://dlmf.nist.gov/18.23.E1
Encodings:: TeX, pMML, png

18.23.2 $\mathop{{{}_{{2}}F_{{0}}}\/}\nolimits\!\left({-x,-x+\beta+N+1\atop-};-z\right)% \*\mathop{{{}_{{2}}F_{{0}}}\/}\nolimits\!\left({x-N,x+\alpha+1\atop-};z\right)% =\sum_{{n=0}}^{N}\frac{\left(-N\right)_{{n}}\left(\alpha+1\right)_{{n}}}{n!}% \mathop{Q_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,N\right)z^{n},$ $x=0,1,\dots,N$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{Q_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,N\right)$ : Hahn polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, : complex variable, : nonnegative integer, : positive integer and : real variable
Referenced by:: §18.23
Permalink:: http://dlmf.nist.gov/18.23.E2
Encodings:: TeX, pMML, png

¶ Krawtchouk

18.23.3 $\left(1-\frac{1-p}{p}z\right)^{x}(1+z)^{{N-x}}=\sum_{{n=0}}^{N}\binom{N}{n}% \mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right)z^{n},$ $x=0,1,\dots,N$ .

Symbols:: $\mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right)$ : Krawtchouk polynomial, $\binom{m}{n}$ : binomial coefficient, : complex variable, : nonnegative integer, : positive integer and : real variable
Referenced by:: §18.23
Permalink:: http://dlmf.nist.gov/18.23.E3
Encodings:: TeX, pMML, png

¶ Meixner

18.23.4 $\left(1-\frac{z}{c}\right)^{x}(1-z)^{{-x-\beta}}=\sum_{{n=0}}^{\infty}\frac{% \left(\beta\right)_{{n}}}{n!}\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c% \right)z^{n},$ $x=0,1,2,\dots$ ,

Symbols:: $\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c\right)$ : Meixner polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, : complex variable, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.23.E4
Encodings:: TeX, pMML, png

¶ Charlier

18.23.5 $e^{z}\left(1-\frac{z}{a}\right)^{x}=\sum_{{n=0}}^{\infty}\frac{\mathop{C_{{n}}% \/}\nolimits\!\left(x,a\right)}{n!}z^{n},$ $x=0,1,2,\dots$ .

Symbols:: $\mathop{C_{{n}}\/}\nolimits\!\left(x,a\right)$ : Charlier polynomial, : base of exponential function, : : factorial, : complex variable, : nonnegative integer and : real variable
Referenced by:: §18.23
Permalink:: http://dlmf.nist.gov/18.23.E5
Encodings:: TeX, pMML, png

¶ Continuous Hahn

18.23.6 $\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left({a+ix\atop 2\realpart{a}};-iz% \right)\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left({\conj{b}-ix\atop 2% \realpart{b}};iz\right)=\sum_{{n=0}}^{\infty}\frac{\mathop{p_{{n}}\/}\nolimits% \!\left(x;a,b,\conj{a},\conj{b}\right)}{\left(2\realpart{a}\right)_{{n}}\left(% 2\realpart{b}\right)_{{n}}}z^{n}.$

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, $\conj{z}$ : complex conjugate, $\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,\conj{a},\conj{b}\right)$ : continuous Hahn polynomial, $\realpart{}$ : real part, : complex variable, : nonnegative integer and : real variable
Referenced by:: §18.23
Permalink:: http://dlmf.nist.gov/18.23.E6
Encodings:: TeX, pMML, png

¶ Meixner–Pollaczek

18.23.7 $(1-e^{{i\phi}}z)^{{-\lambda+ix}}(1-e^{{-i\phi}}z)^{{-\lambda-ix}}=\sum_{{n=0}}% ^{\infty}\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right)z^{n},$

Symbols:: $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right)$ : Meixner–Pollaczek polynomial, : base of exponential function, : complex variable, : nonnegative integer and : real variable
Referenced by:: §18.23
Permalink:: http://dlmf.nist.gov/18.23.E7
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 18.22 Hahn Class: Recurrence Relations and Differences 18.24 Hahn Class: Asymptotic Approximations