# §18.19 Hahn Class: Definitions

## Hahn, Krawtchouk, Meixner, and Charlier

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 $C_{n}\left(x;a\right)$.

## Continuous Hahn

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

 18.19.1 $p_{n}(x)=p_{n}\left(x;a,b,\overline{a},\overline{b}\right),$
 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, $w(x)$: 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 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, $w(x)$: 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 18
 18.19.4 $h_{n}=\frac{2\pi\Gamma\left(n+a+\overline{a}\right)\Gamma\left(n+b+\overline{b% }\right)|\Gamma\left(n+a+\overline{b}\right)|^{2}}{\left(2n+2\Re(a+b)-1\right)% \Gamma\left(n+2\Re(a+b)-1\right)n!},$
 18.19.5 $k_{n}=\frac{{\left(n+2\Re(a+b)-1\right)_{n}}}{n!}.$

## Meixner–Pollaczek

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

 18.19.6 $p_{n}(x)=P^{(\lambda)}_{n}\left(x;\phi\right),$
 18.19.7 $w^{(\lambda)}(z;\phi)=\Gamma\left(\lambda+iz\right)\Gamma\left(\lambda-iz% \right)e^{(2\phi-\pi)z},$
 18.19.8 $w(x)=w^{(\lambda)}(x;\phi)=\left|\Gamma\left(\lambda+\mathrm{i}x\right)\right|% ^{2}e^{(2\phi-\pi)x},$ $\lambda>0$, $0<\phi<\pi$,
 18.19.9 $\displaystyle h_{n}$ $\displaystyle=\frac{2\pi\Gamma\left(n+2\lambda\right)}{(2\sin\phi)^{2\lambda}n% !},$ $\displaystyle k_{n}$ $\displaystyle=\frac{(2\sin\phi)^{n}}{n!}.$