18 Orthogonal Polynomials Askey Scheme 18.19 Hahn Class: Definitions 18.21 Hahn Class: Interrelations

§18.20 Hahn Class: Explicit Representations

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§18.20(i) Rodrigues Formulas
§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

§18.20(i) Rodrigues Formulas

Notes:: For (18.20.2) see Karlin and McGregor (1961, (1.8)). For Table 18.20.1, Rows 2, 3, see Ismail (2005, (6.2.42), (6.1.17)); for Row 4 see Chihara (1978, Chapter V, (3.2)). (18.20.3) follows by iteration of (18.22.28). (18.20.4) follows by iteration of (18.22.30).
Keywords:: Hahn class orthogonal polynomials, Rodrigues formulas
Referenced by:: §18.2(ii), §18.2(ii)
Permalink:: http://dlmf.nist.gov/18.20.i

For comments on the use of the forward-difference operator $\Delta_{{x}}$ , the backward-difference operator $\nabla_{{x}}$ , and the central-difference operator $\delta_{{x}}$ , see §18.2(ii).

¶ Hahn, Krawtchouk, Meixner, and Charlier

18.20.1 $p_{n}(x)=\frac{1}{\kappa_{n}w_{x}}\nabla_{{x}}^{n}\left(w_{x}\prod_{{\ell=0}}^% {{n-1}}F(x+\ell)\right),$ $x\in X$ .

Symbols:: $\nabla_{{x}}$ : backward difference, $\in$ : element of, $w_{x}$ : weights, : subset of $\Real$ , $\ell$ : nonnegative integer, : nonnegative integer, $p_{n}(x)$ : polynomial of degree , : real variable, : coefficient and $\kappa_{n}$ : coefficient
A&S Ref:: 22.17.2
Referenced by:: ¶ ‣ §18.20(i), Table 18.20.1
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In (18.20.1) and $w_{x}$ are as in Table 18.19.1. For the Hahn polynomials $p_{n}(x)=\mathop{Q_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,N\right)$ and

18.20.2

$F(x)=(x+\alpha+1)(x-N),$

$\kappa_{n}=\left(-N\right)_{{n}}\left(\alpha+1\right)_{{n}}.$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : nonnegative integer, : positive integer, : real variable, : coefficient and $\kappa_{n}$ : coefficient
Referenced by:: §18.20(i)
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For the Krawtchouk, Meixner, and Charlier polynomials, and $\kappa_{n}$ are as in Table 18.20.1.

Table 18.20.1: Krawtchouk, Meixner, and Charlier OP’s: Rodrigues formulas (18.20.1).

$p_{n}(x)$		$\kappa_{n}$
$\mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right)$		$\left(-N\right)_{{n}}$
$\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c\right)$	$x+\beta$	$\left(\beta\right)_{{n}}$
$\mathop{C_{{n}}\/}\nolimits\!\left(x,a\right)$	1	1

Symbols:: $\mathop{C_{{n}}\/}\nolimits\!\left(x,a\right)$ : Charlier polynomial, $\mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right)$ : Krawtchouk polynomial, $\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c\right)$ : Meixner polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : nonnegative integer, : positive integer, $p_{n}(x)$ : polynomial of degree , : real variable, : coefficient and $\kappa_{n}$ : coefficient
Referenced by:: ¶ ‣ §18.20(i), §18.20(i)
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¶ Continuous Hahn

18.20.3 $w(x;a,b,\conj{a},\conj{b})\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,\conj{a},% \conj{b}\right)=\frac{1}{n!}\delta_{{x}}^{n}\left(w(x;a+\tfrac{1}{2}n,b+\tfrac% {1}{2}n,\conj{a}+\tfrac{1}{2}n,\conj{b}+\tfrac{1}{2}n)\right).$

Symbols:: $\delta_{{x}}$ : central difference, $\conj{z}$ : complex conjugate, $\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,\conj{a},\conj{b}\right)$ : continuous Hahn polynomial, : : factorial, : nonnegative integer and : real variable
Referenced by:: §18.20(i)
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¶ Meixner–Pollaczek

18.20.4 $w^{{(\lambda)}}(x;\phi)\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi% \right)=\frac{1}{n!}\delta_{{x}}^{n}\left(w^{{(\lambda+\frac{1}{2}n)}}(x;\phi)% \right).$

Symbols:: $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right)$ : Meixner–Pollaczek polynomial, $\delta_{{x}}$ : central difference, : : factorial, : nonnegative integer and : real variable
Referenced by:: §18.20(i)
Permalink:: http://dlmf.nist.gov/18.20.E4
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§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

Notes:: For (18.20.5)–(18.20.8) and (18.20.10) see Ismail (2005, (6.2.3), (6.2.34), (6.1.3), (6.1.20), (5.9.5)). For (18.20.9) see Askey (1985).
Keywords:: Hahn class orthogonal polynomials, generalized hypergeometric functions, hypergeometric function
Referenced by:: §15.9(i)
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For the definition of hypergeometric and generalized hypergeometric functions see §16.2.

18.20.5 $\mathop{Q_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,N\right)=\mathop{{{}_{{3}}F_% {{2}}}\/}\nolimits\!\left({-n,n+\alpha+\beta+1,-x\atop\alpha+1,-N};1\right),$ $n=0,1,\dots,N$ .

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, : nonnegative integer, : positive integer and : real variable
Referenced by:: §18.20(ii), §18.21(ii), §18.23, §18.26(ii)
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18.20.6 $\mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right)=\mathop{{{}_{{2}}F_{{1}}}\/}% \nolimits\!\left({-n,-x\atop-N};p^{{-1}}\right),$ $n=0,1,\dots,N$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right)$ : Krawtchouk polynomial, : nonnegative integer, : positive integer and : real variable
Referenced by:: §18.19, §18.21(ii)
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18.20.7 $\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c\right)=\mathop{{{}_{{2}}F_{{1}}}% \/}\nolimits\!\left({-n,-x\atop\beta};1-c^{{-1}}\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c\right)$ : Meixner polynomial, : nonnegative integer and : real variable
Referenced by:: §18.21(ii), §18.26(ii)
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18.20.8 $\mathop{C_{{n}}\/}\nolimits\!\left(x,a\right)=\mathop{{{}_{{2}}F_{{0}}}\/}% \nolimits\!\left({-n,-x\atop-};-a^{{-1}}\right).$

Symbols:: $\mathop{C_{{n}}\/}\nolimits\!\left(x,a\right)$ : Charlier polynomial, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, : nonnegative integer and : real variable
Referenced by:: §13.6(v), §18.19, §18.20(ii), §18.21(ii)
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18.20.9 $\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,\conj{a},\conj{b}\right)=\frac{i^{n}% \left(a+\conj{a}\right)_{{n}}\left(a+\conj{b}\right)_{{n}}}{n!}\*\mathop{{{}_{% {3}}F_{{2}}}\/}\nolimits\!\left({-n,n+2\realpart{(a+b)}-1,a+ix\atop a+\conj{a}% ,a+\conj{b}};1\right).$

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, : : factorial, $\realpart{}$ : real part, : nonnegative integer and : real variable
Referenced by:: §18.19, §18.20(ii), §18.21(ii), §18.22(iii), §18.26(ii)
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(For symmetry properties of $\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,\conj{a},\conj{b}\right)$ with respect to , , $\conj{a}$ , $\conj{b}$ see Andrews et al. (1999, Corollary 3.3.4).)

18.20.10 $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right)=\frac{\left(2% \lambda\right)_{{n}}}{n!}e^{{in\phi}}\*\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!% \left({-n,\lambda+ix\atop 2\lambda};1-e^{{-2i\phi}}\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right)$ : Meixner–Pollaczek polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : base of exponential function, : : factorial, : nonnegative integer and : real variable
Referenced by:: §18.19, §18.20(ii), §18.21(ii), §18.22(iii), §18.26(ii)
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© 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. 18.19 Hahn Class: Definitions 18.21 Hahn Class: Interrelations