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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Permalink:: http://dlmf.nist.gov/18.20
See also:: Annotations for Ch.18

§18.20(i) Rodrigues Formulas

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Keywords:: Hahn class orthogonal polynomials, Rodrigues formula, Rodrigues formulas
Notes:: For (18.20.2) see Karlin and McGregor (1961, (1.8)). For Table 18.20.1, Rows 2, 3, see Ismail (2009, (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).
Referenced by:: §18.2(ii), §18.2(ii)
Permalink:: http://dlmf.nist.gov/18.20.i
See also:: Annotations for §18.20 and Ch.18

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

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See also:: Annotations for §18.20(i), §18.20 and Ch.18

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: $\nabla_{\NVar{x}}$ : backward difference, $\in$ : element of, $p_{n}(x)$ : polynomial of degree $n$ , $w_{x}$ : weights, $X$ : subset of $\mathbb{R}$ , $\ell$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $F(x)$ : coefficient and $\kappa_{n}$ : coefficient A&S Ref: 22.17.2 Referenced by: §18.20(i), Table 18.20.1 Permalink: http://dlmf.nist.gov/18.20.E1 Encodings: TeX, pMML, png See also: Annotations for §18.20(i), §18.20(i), §18.20 and Ch.18

In (18.20.1) $X$ and $w_{x}$ are as in Table 18.19.1. For the Hahn polynomials $p_{n}(x)=Q_{n}\left(x;\alpha,\beta,N\right)$ and

18.20.2	$\displaystyle F(x)$	$\displaystyle=(x+\alpha+1)(x-N),$
18.20.2	$\displaystyle\kappa_{n}$	$\displaystyle={\left(-N\right)_{n}}{\left(\alpha+1\right)_{n}}.$
ⓘ Defines: $F(x)$ : coefficient (locally) and $\kappa_{n}$ : coefficient (locally) Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $N$ : positive integer, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.20(i) Permalink: http://dlmf.nist.gov/18.20.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.20(i), §18.20(i), §18.20 and Ch.18

For the Krawtchouk, Meixner, and Charlier polynomials, $F(x)$ 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).

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

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Symbols:: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $K_{\NVar{n}}\left(\NVar{x};\NVar{p},\NVar{N}\right)$ : Krawtchouk polynomial, $M_{\NVar{n}}\left(\NVar{x};\NVar{\beta},\NVar{c}\right)$ : Meixner polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $N$ : positive integer, $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer, $x$ : real variable, $F(x)$ : coefficient and $\kappa_{n}$ : coefficient
Referenced by:: §18.20(i), §18.20(i)
Permalink:: http://dlmf.nist.gov/18.20.T1
See also:: Annotations for §18.20(i), §18.20(i), §18.20 and Ch.18

Continuous Hahn

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See also:: Annotations for §18.20(i), §18.20 and Ch.18

18.20.3		$w(x;a,b,\overline{a},\overline{b})p_{n}\left(x;a,b,\overline{a},\overline{b}% \right)=\frac{1}{n!}\delta_{x}^{n}\left(w(x;a+\tfrac{1}{2}n,b+\tfrac{1}{2}n,% \overline{a}+\tfrac{1}{2}n,\overline{b}+\tfrac{1}{2}n)\right).$
ⓘ Symbols: $\delta_{\NVar{x}}$ : central difference, $\overline{\NVar{z}}$ : complex conjugate, $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{\overline{a}},\NVar{% \overline{b}}\right)$ : continuous Hahn polynomial, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.20(i) Permalink: http://dlmf.nist.gov/18.20.E3 Encodings: TeX, pMML, png See also: Annotations for §18.20(i), §18.20(i), §18.20 and Ch.18

Meixner–Pollaczek

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See also:: Annotations for §18.20(i), §18.20 and Ch.18

18.20.4		$w^{(\lambda)}(x;\phi)P^{(\lambda)}_{n}\left(x;\phi\right)=\frac{1}{n!}\delta_{% x}^{n}\left(w^{(\lambda+\frac{1}{2}n)}(x;\phi)\right).$
ⓘ Symbols: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$ : Meixner–Pollaczek polynomial, $\delta_{\NVar{x}}$ : central difference, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.20(i) Permalink: http://dlmf.nist.gov/18.20.E4 Encodings: TeX, pMML, png See also: Annotations for §18.20(i), §18.20(i), §18.20 and Ch.18

§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

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Keywords:: Hahn class orthogonal polynomials, generalized hypergeometric functions, hypergeometric function, relations to other functions
Notes:: For (18.20.5)–(18.20.8) and (18.20.10) see Ismail (2009, (6.2.3), (6.2.34), (6.1.3), (6.1.20), (5.9.5)). For (18.20.9) see Askey (1985).
Referenced by:: §15.9(i)
Permalink:: http://dlmf.nist.gov/18.20.ii
Addition (effective with 1.2.0):: A sentence was added just before (18.20.5) giving some conventions.
See also:: Annotations for §18.20 and Ch.18

For the definition of hypergeometric and generalized hypergeometric functions see §16.2. Here we use as convention for (16.2.1) with $b_{q}=-N$ , $a_{1}=-n$ , and $n=0,1,\ldots,N$ that the summation on the right-hand side ends at $k=n$ .

18.20.5		$Q_{n}\left(x;\alpha,\beta,N\right)={{}_{3}F_{2}}\left({-n,n+\alpha+\beta+1,-x% \atop\alpha+1,-N};1\right),$
$n=0,1,\dots,N$ .
ⓘ 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, $N$ : positive integer, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.20(ii), §18.21(ii), §18.23, §18.26(ii), §18.27(ii), §18.38(iii) Permalink: http://dlmf.nist.gov/18.20.E5 Encodings: TeX, pMML, png See also: Annotations for §18.20(ii), §18.20 and Ch.18

18.20.6	$\displaystyle K_{n}\left(x;p,N\right)$	$\displaystyle={{}_{2}F_{1}}\left({-n,-x\atop-N};p^{-1}\right),$
$n=0,1,\dots,N$ .
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $K_{\NVar{n}}\left(\NVar{x};\NVar{p},\NVar{N}\right)$ : Krawtchouk polynomial, $N$ : positive integer, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.19, §18.21(ii) Permalink: http://dlmf.nist.gov/18.20.E6 Encodings: TeX, pMML, png See also: Annotations for §18.20(ii), §18.20 and Ch.18
18.20.7	$\displaystyle M_{n}\left(x;\beta,c\right)$	$\displaystyle={{}_{2}F_{1}}\left({-n,-x\atop\beta};1-c^{-1}\right).$
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $M_{\NVar{n}}\left(\NVar{x};\NVar{\beta},\NVar{c}\right)$ : Meixner polynomial, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.21(ii), §18.26(ii) Permalink: http://dlmf.nist.gov/18.20.E7 Encodings: TeX, pMML, png See also: Annotations for §18.20(ii), §18.20 and Ch.18
18.20.8	$\displaystyle C_{n}\left(x;a\right)$	$\displaystyle={{}_{2}F_{0}}\left({-n,-x\atop-};-a^{-1}\right).$
ⓘ Symbols: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $n$ : nonnegative integer and $x$ : real variable Referenced by: §13.6(v), §18.19, §18.20(ii), §18.21(ii) Permalink: http://dlmf.nist.gov/18.20.E8 Encodings: TeX, pMML, png See also: Annotations for §18.20(ii), §18.20 and Ch.18

18.20.9		$p_{n}\left(x;a,b,\overline{a},\overline{b}\right)=\frac{{\mathrm{i}}^{n}{\left% (a+\overline{a}\right)_{n}}{\left(a+\overline{b}\right)_{n}}}{n!}\*{{}_{3}F_{2% }}\left({-n,n+2\Re\left(a+b\right)-1,a+\mathrm{i}x\atop a+\overline{a},a+% \overline{b}};1\right).$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $\Re$ : real part, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.19, §18.20(ii), §18.21(ii), §18.22(iii), §18.26(ii) Permalink: http://dlmf.nist.gov/18.20.E9 Encodings: TeX, pMML, png See also: Annotations for §18.20(ii), §18.20 and Ch.18

(For symmetry properties of $p_{n}\left(x;a,b,\overline{a},\overline{b}\right)$ with respect to $a$ , $b$ , $\overline{a}$ , $\overline{b}$ see Andrews et al. (1999, Corollary 3.3.4).)

18.20.10		$P^{(\lambda)}_{n}\left(x;\phi\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}{% \mathrm{e}}^{\mathrm{i}n\phi}\*{{}_{2}F_{1}}\left({-n,\lambda+\mathrm{i}x\atop 2% \lambda};1-{\mathrm{e}}^{-2\mathrm{i}\phi}\right).$
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$ : Meixner–Pollaczek polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.19, §18.20(ii), §18.21(ii), §18.22(iii), §18.26(ii), (18.35.9) Permalink: http://dlmf.nist.gov/18.20.E10 Encodings: TeX, pMML, png See also: Annotations for §18.20(ii), §18.20 and Ch.18

18.19 Hahn Class: Definitions 18.21 Hahn Class: Interrelations

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