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6000 DLMF: §18.23 Hahn Class: Generating Functions ‣ Askey Scheme ‣ Chapter 18 Orthogonal Polynomials

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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

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Keywords:: Hahn class orthogonal polynomials, generating functions
Notes:: For (18.23.3)–(18.23.5), (18.23.7) see Ismail (2009, (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 $z$ , and substituting (18.20.5) on the right. (18.23.6) is a limiting case of (18.26.18) via (18.26.6).
Permalink:: http://dlmf.nist.gov/18.23
See also:: Annotations for Ch.18

For the definition of generalized hypergeometric functions see §16.2.

Hahn

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Keywords:: Hahn class orthogonal polynomials, confluent hypergeometric functions, relations to other functions
See also:: Annotations for §18.23 and Ch.18

18.23.1		${{}_{1}F_{1}}\left({-x\atop\alpha+1};-z\right){{}_{1}F_{1}}\left({x-N\atop% \beta+1};z\right)=\sum_{n=0}^{N}\frac{{\left(-N\right)_{n}}}{{\left(\beta+1% \right)_{n}}n!}Q_{n}\left(x;\alpha,\beta,N\right)z^{n},$
$x=0,1,\dots,N$ .
ⓘ Symbols: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, ${{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ : $=M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $N$ : positive integer, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.23 Permalink: http://dlmf.nist.gov/18.23.E1 Encodings: TeX, pMML, png See also: Annotations for §18.23, §18.23 and Ch.18

18.23.2		${{}_{2}F_{0}}\left({-x,-x+\beta+N+1\atop-};-z\right)\*{{}_{2}F_{0}}\left({x-N,% x+\alpha+1\atop-};z\right)=\sum_{n=0}^{N}\frac{{\left(-N\right)_{n}}{\left(% \alpha+1\right)_{n}}}{n!}Q_{n}\left(x;\alpha,\beta,N\right)z^{n},$
$x=0,1,\dots,N$ .
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized hypergeometric function, $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $N$ : positive integer, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.23 Permalink: http://dlmf.nist.gov/18.23.E2 Encodings: TeX, pMML, png See also: Annotations for §18.23, §18.23 and Ch.18

Krawtchouk

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See also:: Annotations for §18.23 and Ch.18

18.23.3		$\left(1-\frac{1-p}{p}z\right)^{x}(1+z)^{N-x}=\sum_{n=0}^{N}\genfrac{(}{)}{0.0% pt}{}{N}{n}K_{n}\left(x;p,N\right)z^{n},$
$x=0,1,\dots,N$ .
ⓘ Symbols: $K_{\NVar{n}}\left(\NVar{x};\NVar{p},\NVar{N}\right)$ : Krawtchouk polynomial, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $N$ : positive integer, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.2(xii), §18.23 Permalink: http://dlmf.nist.gov/18.23.E3 Encodings: TeX, pMML, png See also: Annotations for §18.23, §18.23 and Ch.18

Meixner

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See also:: Annotations for §18.23 and Ch.18

18.23.4		$\left(1-\frac{z}{c}\right)^{x}(1-z)^{-x-\beta}=\sum_{n=0}^{\infty}\frac{{\left% (\beta\right)_{n}}}{n!}M_{n}\left(x;\beta,c\right)z^{n},$
$x=0,1,2,\dots$ , $\|z\|<1$ .
ⓘ Symbols: $M_{\NVar{n}}\left(\NVar{x};\NVar{\beta},\NVar{c}\right)$ : Meixner polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.2(xii) Permalink: http://dlmf.nist.gov/18.23.E4 Encodings: TeX, pMML, png See also: Annotations for §18.23, §18.23 and Ch.18

Charlier

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See also:: Annotations for §18.23 and Ch.18

18.23.5		${\mathrm{e}}^{z}\left(1-\frac{z}{a}\right)^{x}=\sum_{n=0}^{\infty}\frac{C_{n}% \left(x;a\right)}{n!}z^{n},$
$x=0,1,2,\dots$ .
ⓘ Symbols: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.2(xii), §18.23 Permalink: http://dlmf.nist.gov/18.23.E5 Encodings: TeX, pMML, png See also: Annotations for §18.23, §18.23 and Ch.18

Continuous Hahn

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See also:: Annotations for §18.23 and Ch.18

18.23.6		${{}_{1}F_{1}}\left({a+\mathrm{i}x\atop 2\Re a};-\mathrm{i}z\right){{}_{1}F_{1}% }\left({\overline{b}-\mathrm{i}x\atop 2\Re b};\mathrm{i}z\right)=\sum_{n=0}^{% \infty}\frac{p_{n}\left(x;a,b,\overline{a},\overline{b}\right)}{{\left(2\Re a% \right)_{n}}{\left(2\Re b\right)_{n}}}z^{n}.$
ⓘ Symbols: ${{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ : $=M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\overline{\NVar{z}}$ : complex conjugate, $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{\overline{a}},\NVar{% \overline{b}}\right)$ : continuous Hahn polynomial, $\mathrm{i}$ : imaginary unit, $\Re$ : real part, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.23 Permalink: http://dlmf.nist.gov/18.23.E6 Encodings: TeX, pMML, png See also: Annotations for §18.23, §18.23 and Ch.18

Meixner–Pollaczek

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See also:: Annotations for §18.23 and Ch.18

18.23.7		$(1-{\mathrm{e}}^{\mathrm{i}\phi}z)^{-\lambda+\mathrm{i}x}(1-{\mathrm{e}}^{-% \mathrm{i}\phi}z)^{-\lambda-\mathrm{i}x}=\sum_{n=0}^{\infty}P^{(\lambda)}_{n}% \left(x;\phi\right)z^{n},$
$\|z\|<1$ .
ⓘ Symbols: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$ : Meixner–Pollaczek polynomial, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.2(xii), §18.23 Permalink: http://dlmf.nist.gov/18.23.E7 Encodings: TeX, pMML, png See also: Annotations for §18.23, §18.23 and Ch.18

18.22 Hahn Class: Recurrence Relations and Differences 18.24 Hahn Class: Asymptotic Approximations

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