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

§18.19 Hahn Class: Definitions

Notes:: For Table 18.19.1 see Ismail (2005, (6.2.4), (6.2.35), (6.1.4), (6.1.21)). For Table 18.19.2, Rows 2, 4, see Ismail (2005, (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 (2005, (5.9.8), (5.9.9)). The formula for $k_{n}$ in (18.19.9) follows from (18.20.10).
Keywords:: Hahn class orthogonal polynomials
Referenced by:: §13.6(v), §15.9(i), §18.35(iii)
Permalink:: http://dlmf.nist.gov/18.19

¶ Hahn, Krawtchouk, Meixner, and Charlier

Keywords:: Hahn class orthogonal polynomials

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

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

$p_{n}(x)$		$w_{x}$	$h_{n}$

$\mathop{Q_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,N\right)$ , $n=0,1,\ldots,N$	$\{0,1,\ldots,N\}$	$\dfrac{\left(\alpha+1\right)_{{x}}\left(\beta+1\right)_{{N-x}}}{x!(N-x)!}$ , $\alpha,\beta>-1$ 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 $(-1)^{N}h_{n}>0$ .


$\mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right)$ , $n=0,1,\dots,N$	$\{0,1,\dots,N\}$	$\dbinom{N}{x}p^{x}(1-p)^{{N-x}}$ ,	$\left(\dfrac{1-p}{p}\right)^{n}\Bigg/\dbinom{N}{n}$


$\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c\right)$	$\{0,1,2,\dots\}$	$\ifrac{\left(\beta\right)_{{x}}c^{x}}{x!}$ , $\beta>0$ ,	$\dfrac{c^{{-n}}n!}{\left(\beta\right)_{{n}}(1-c)^{{\beta}}}$

$\mathop{C_{{n}}\/}\nolimits\!\left(x,a\right)$	$\{0,1,2,\dots\}$	$\ifrac{a^{x}}{x!}$ ,	$a^{{-n}}e^{a}n!$

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

$p_{n}(x)$	$k_{n}$

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

Symbols:: $\mathop{C_{{n}}\/}\nolimits\!\left(x,a\right)$ : Charlier polynomial, $\mathop{Q_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,N\right)$ : Hahn 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 and $k_{n}$
Keywords:: Hahn class orthogonal polynomials
Referenced by:: ¶ ‣ §18.19, §18.19
Permalink:: http://dlmf.nist.gov/18.19.T2

¶ Continuous Hahn

Defines:: $\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,\conj{a},\conj{b}\right)$ : continuous Hahn polynomial

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

18.19.1 $p_{n}(x)=\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,\conj{a},\conj{b}\right),$

Symbols:: $\conj{z}$ : complex conjugate, $\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,\conj{a},\conj{b}\right)$ : continuous Hahn polynomial, : nonnegative integer, $p_{n}(x)$ : polynomial of degree , : complex, : complex and : real variable
Referenced by:: §18.19
Permalink:: http://dlmf.nist.gov/18.19.E1
Encodings:: TeX, pMML, png

18.19.2 $w(z;a,b,\conj{a},\conj{b})=\mathop{\Gamma\/}\nolimits\!\left(a+iz\right)% \mathop{\Gamma\/}\nolimits\!\left(b+iz\right)\mathop{\Gamma\/}\nolimits\!\left% (\conj{a}-iz\right)\mathop{\Gamma\/}\nolimits\!\left(\conj{b}-iz\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\conj{z}$ : complex conjugate, : weight function, : complex variable, : complex and : complex
Permalink:: http://dlmf.nist.gov/18.19.E2
Encodings:: TeX, pMML, png

18.19.3 $w(x)=w(x;a,b,\conj{a},\conj{b})=|\mathop{\Gamma\/}\nolimits\!\left(a+ix\right)% \mathop{\Gamma\/}\nolimits\!\left(b+ix\right)|^{2},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\conj{z}$ : complex conjugate, : weight function, : complex, : complex and : real variable
Permalink:: http://dlmf.nist.gov/18.19.E3
Encodings:: TeX, pMML, png

18.19.4 $h_{n}=\frac{2\pi\mathop{\Gamma\/}\nolimits\!\left(n+a+\conj{a}\right)\mathop{% \Gamma\/}\nolimits\!\left(n+b+\conj{b}\right)|\mathop{\Gamma\/}\nolimits\!% \left(n+a+\conj{b}\right)|^{2}}{\left(2n+2\realpart{(a+b)}-1\right)\mathop{% \Gamma\/}\nolimits\!\left(n+2\realpart{(a+b)}-1\right)n!},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\conj{z}$ : complex conjugate, : : factorial, $\realpart{}$ : real part, : nonnegative integer, : complex, : complex and $h_{n}$
Referenced by:: §18.19
Permalink:: http://dlmf.nist.gov/18.19.E4
Encodings:: TeX, pMML, png

18.19.5 $k_{n}=\frac{\left(n+2\realpart{(a+b)}-1\right)_{{n}}}{n!}.$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, $\realpart{}$ : real part, : nonnegative integer, : complex, : complex and $k_{n}$
Referenced by:: §18.19
Permalink:: http://dlmf.nist.gov/18.19.E5
Encodings:: TeX, pMML, png

¶ Meixner–Pollaczek

Defines:: $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right)$ : Meixner–Pollaczek polynomial
Keywords:: Hahn class orthogonal polynomials

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

18.19.6 $p_{n}(x)=\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right),$

Symbols:: $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right)$ : Meixner–Pollaczek polynomial, : nonnegative integer, $p_{n}(x)$ : polynomial of degree and : real variable
Referenced by:: §18.19
Permalink:: http://dlmf.nist.gov/18.19.E6
Encodings:: TeX, pMML, png

18.19.7 $w^{{(\lambda)}}(z;\phi)=\mathop{\Gamma\/}\nolimits\!\left(\lambda+iz\right)% \mathop{\Gamma\/}\nolimits\!\left(\lambda-iz\right)e^{{(2\phi-\pi)z}},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : base of exponential function, : weight function and : complex variable
Permalink:: http://dlmf.nist.gov/18.19.E7
Encodings:: TeX, pMML, png

18.19.8 $w(x)=w^{{(\lambda)}}(x;\phi)=\left|\mathop{\Gamma\/}\nolimits\!\left(\lambda+% ix\right)\right|^{2}e^{{(2\phi-\pi)x}},$ $\lambda>0$ , $0<\phi<\pi$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : base of exponential function, : weight function and : real variable
Permalink:: http://dlmf.nist.gov/18.19.E8
Encodings:: TeX, pMML, png

18.19.9

$h_{n}=\frac{2\pi\mathop{\Gamma\/}\nolimits\!\left(n+2\lambda\right)}{(2\mathop% {\sin\/}\nolimits\phi)^{{2\lambda}}n!},$

$k_{n}=\frac{(2\mathop{\sin\/}\nolimits\phi)^{n}}{n!}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : : factorial, $\mathop{\sin\/}\nolimits z$ : sine function, : 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

© 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. 18.18 Sums 18.20 Hahn Class: Explicit Representations