18 Orthogonal Polynomials Askey Scheme 18.18 Sums 18.20 Hahn Class: Explicit Representations

§18.19 Hahn Class: Definitions

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Keywords:: Hahn class orthogonal polynomials, definitions
Notes:: For Table 18.19.1 see Ismail (2009, (6.2.4), (6.2.35), (6.1.4), (6.1.21)). For Table 18.19.2, Rows 2, 4, see Ismail (2009, (6.2.7), (6.1.7)); Row 3 follows from (18.20.6); Row 5 follows from (18.20.8). For (18.19.1)–(18.19.4) see Askey (1985, (4), (5)). (18.19.5) follows from (18.20.9). For (18.19.6)–(18.19.9) see Ismail (2009, (5.9.8), (5.9.9)). The formula for $k_{n}$ in (18.19.9) follows from (18.20.10).
Referenced by:: §13.6(v), §15.9(i), §18.35(iii)
Permalink:: http://dlmf.nist.gov/18.19
See also:: Annotations for Ch.18

Hahn, Krawtchouk, Meixner, and Charlier

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Keywords:: Hahn class orthogonal polynomials, orthogonality properties
See also:: Annotations for §18.19 and Ch.18

Tables 18.19.1 and 18.19.2 provide definitions via orthogonality and normalization (§§18.2(i), 18.2(iii)) for the Hahn polynomials $Q_{n}\left(x;\alpha,\beta,N\right)$ , Krawtchouk polynomials $K_{n}\left(x;p,N\right)$ , Meixner polynomials $M_{n}\left(x;\beta,c\right)$ , and Charlier polynomials $C_{n}\left(x;a\right)$ .

Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, normalizations, and parameter constraints.

$p_{n}(x)$	$X$	$w_{x}$	$h_{n}$
$Q_{n}\left(x;\alpha,\beta,N\right)$ , $n=0,1,\ldots,N$	$\{0,1,\ldots,N\}$	$\dfrac{{\left(\alpha+1\right)_{x}}{\left(\beta+1\right)_{N-x}}}{x!(N-x)!}$ , $\alpha,\beta>-1$ or $\alpha,\beta<-N$	$\dfrac{(-1)^{n}{\left(n+\alpha+\beta+1\right)_{N+1}}{\left(\beta+1\right)_{n}}% n!}{(2n+\alpha+\beta+1){\left(\alpha+1\right)_{n}}{\left(-N\right)_{n}}N!}$ If $\alpha,\beta<-N$ , then $(-1)^{N}w_{x}>0$ and $(-1)^{N}h_{n}>0$ .
$K_{n}\left(x;p,N\right)$ , $n=0,1,\dots,N$	$\{0,1,\dots,N\}$	$\dbinom{N}{x}p^{x}(1-p)^{N-x}$ , $0<p<1$	$\left(\dfrac{1-p}{p}\right)^{n}\Bigg{/}\dbinom{N}{n}$
$M_{n}\left(x;\beta,c\right)$	$\{0,1,2,\dots\}$	$\ifrac{{\left(\beta\right)_{x}}c^{x}}{x!}$ , $\beta>0$ , $0<c<1$	$\dfrac{c^{-n}n!}{{\left(\beta\right)_{n}}(1-c)^{\beta}}$
$C_{n}\left(x;a\right)$	$\{0,1,2,\dots\}$	$\ifrac{a^{x}}{x!}$ , $a>0$	$a^{-n}e^{a}n!$

Table 18.19.2: Hahn, Krawtchouk, Meixner, and Charlier OP’s: leading coefficients.

$p_{n}(x)$	$k_{n}$
$Q_{n}\left(x;\alpha,\beta,N\right)$	$\dfrac{{\left(n+\alpha+\beta+1\right)_{n}}}{{\left(\alpha+1\right)_{n}}{\left(% -N\right)_{n}}}$
$K_{n}\left(x;p,N\right)$	$\ifrac{p^{-n}}{{\left(-N\right)_{n}}}$
$M_{n}\left(x;\beta,c\right)$	$\ifrac{(1-c^{-1})^{n}}{{\left(\beta\right)_{n}}}$
$C_{n}\left(x;a\right)$	$(-a)^{-n}$

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Symbols:: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn 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), $n$ : nonnegative integer, $N$ : positive integer, $p_{n}(x)$ : polynomial of degree $n$ , $x$ : real variable and $k_{n}$
Keywords:: Hahn class orthogonal polynomials, leading coefficients
Referenced by:: §18.19, §18.19
Permalink:: http://dlmf.nist.gov/18.19.T2
See also:: Annotations for §18.19, §18.19 and Ch.18

Continuous Hahn

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Defines:: $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{\overline{a}},\NVar{% \overline{b}}\right)$ : continuous Hahn polynomial
See also:: Annotations for §18.19 and Ch.18

These polynomials are orthogonal on $(-\infty,\infty)$ , and with $\Re a>0$ , $\Re b>0$ are defined as follows.

18.19.1		$p_{n}(x)=p_{n}\left(x;a,b,\overline{a},\overline{b}\right),$
ⓘ Symbols: $\overline{\NVar{z}}$ : complex conjugate, $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{\overline{a}},\NVar{% \overline{b}}\right)$ : continuous Hahn polynomial, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable Referenced by: §18.19 Permalink: http://dlmf.nist.gov/18.19.E1 Encodings: TeX, pMML, png See also: Annotations for §18.19, §18.19 and Ch.18

18.19.2		$w(z;a,b,\overline{a},\overline{b})=\Gamma\left(a+iz\right)\Gamma\left(b+iz% \right)\Gamma\left(\overline{a}-iz\right)\Gamma\left(\overline{b}-iz\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\overline{\NVar{z}}$ : complex conjugate, $\mathrm{i}$ : imaginary unit, $w(x)$ : weight function and $z$ : complex variable Permalink: http://dlmf.nist.gov/18.19.E2 Encodings: TeX, pMML, png See also: Annotations for §18.19, §18.19 and Ch.18

18.19.3		$w(x)=w(x;a,b,\overline{a},\overline{b})=\|\Gamma\left(a+\mathrm{i}x\right)% \Gamma\left(b+\mathrm{i}x\right)\|^{2},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\overline{\NVar{z}}$ : complex conjugate, $\mathrm{i}$ : imaginary unit, $w(x)$ : weight function and $x$ : real variable Permalink: http://dlmf.nist.gov/18.19.E3 Encodings: TeX, pMML, png See also: Annotations for §18.19, §18.19 and Ch.18

18.19.4		$h_{n}=\frac{2\pi\Gamma\left(n+a+\overline{a}\right)\Gamma\left(n+b+\overline{b% }\right)\|\Gamma\left(n+a+\overline{b}\right)\|^{2}}{\left(2n+2\Re\left(a+b% \right)-1\right)\Gamma\left(n+2\Re\left(a+b\right)-1\right)n!},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $!$ : factorial (as in $n!$ ), $\Re$ : real part, $n$ : nonnegative integer and $h_{n}$ Referenced by: §18.19 Permalink: http://dlmf.nist.gov/18.19.E4 Encodings: TeX, pMML, png See also: Annotations for §18.19, §18.19 and Ch.18

18.19.5		$k_{n}=\frac{{\left(n+2\Re\left(a+b\right)-1\right)_{n}}}{n!}.$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\Re$ : real part, $n$ : nonnegative integer and $k_{n}$ Referenced by: §18.19 Permalink: http://dlmf.nist.gov/18.19.E5 Encodings: TeX, pMML, png See also: Annotations for §18.19, §18.19 and Ch.18

Meixner–Pollaczek

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Defines:: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$ : Meixner–Pollaczek polynomial
Keywords:: Hahn class orthogonal polynomials, definitions, orthogonality properties
See also:: Annotations for §18.19 and Ch.18

These polynomials are orthogonal on $(-\infty,\infty)$ , and are defined as follows.

18.19.6		$p_{n}(x)=P^{(\lambda)}_{n}\left(x;\phi\right),$
ⓘ Symbols: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$ : Meixner–Pollaczek polynomial, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable Referenced by: §18.19 Permalink: http://dlmf.nist.gov/18.19.E6 Encodings: TeX, pMML, png See also: Annotations for §18.19, §18.19 and Ch.18

18.19.7		$w^{(\lambda)}(z;\phi)=\Gamma\left(\lambda+iz\right)\Gamma\left(\lambda-iz% \right)e^{(2\phi-\pi)z},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $w(x)$ : weight function and $z$ : complex variable Permalink: http://dlmf.nist.gov/18.19.E7 Encodings: TeX, pMML, png See also: Annotations for §18.19, §18.19 and Ch.18

18.19.8		$w(x)=w^{(\lambda)}(x;\phi)=\left\|\Gamma\left(\lambda+\mathrm{i}x\right)\right\|% ^{2}e^{(2\phi-\pi)x},$
$\lambda>0$ , $0<\phi<\pi$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $w(x)$ : weight function and $x$ : real variable Permalink: http://dlmf.nist.gov/18.19.E8 Encodings: TeX, pMML, png See also: Annotations for §18.19, §18.19 and Ch.18

18.19.9	$\displaystyle h_{n}$	$\displaystyle=\frac{2\pi\Gamma\left(n+2\lambda\right)}{(2\sin\phi)^{2\lambda}n% !},$
18.19.9	$\displaystyle k_{n}$	$\displaystyle=\frac{(2\sin\phi)^{n}}{n!}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function, $n$ : nonnegative integer, $h_{n}$ and $k_{n}$ Referenced by: §18.19 Permalink: http://dlmf.nist.gov/18.19.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.19, §18.19 and Ch.18