# 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 $\mathop{Q_{n}\/}\nolimits\!\left(x;\alpha,\beta,N\right)$, 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 $\mathop{C_{n}\/}\nolimits\!\left(x,a\right)$.

# Continuous Hahn

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

 18.19.1 $p_{n}(x)=\mathop{p_{n}\/}\nolimits\!\left(x;a,b,\conj{a},\conj{b}\right),$
 18.19.2 $w(z;a,b,\conj{a},\conj{b})=\mathop{\Gamma\/}\nolimits\!\left(a+iz\right)% \mathop{\Gamma\/}\nolimits\!\left(b+iz\right)\mathop{\Gamma\/}\nolimits\!\left% (\conj{a}-iz\right)\mathop{\Gamma\/}\nolimits\!\left(\conj{b}-iz\right),$
 18.19.3 $w(x)=w(x;a,b,\conj{a},\conj{b})=|\mathop{\Gamma\/}\nolimits\!\left(a+ix\right)% \mathop{\Gamma\/}\nolimits\!\left(b+ix\right)|^{2},$
 18.19.4 $h_{n}=\frac{2\pi\mathop{\Gamma\/}\nolimits\!\left(n+a+\conj{a}\right)\mathop{% \Gamma\/}\nolimits\!\left(n+b+\conj{b}\right)|\mathop{\Gamma\/}\nolimits\!% \left(n+a+\conj{b}\right)|^{2}}{\left(2n+2\realpart{(a+b)}-1\right)\mathop{% \Gamma\/}\nolimits\!\left(n+2\realpart{(a+b)}-1\right)n!},$
 18.19.5 $k_{n}=\frac{\left(n+2\realpart{(a+b)}-1\right)_{n}}{n!}.$

# Meixner–Pollaczek

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

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