# §18.23 Hahn Class: Generating Functions

For the definition of generalized hypergeometric functions see §16.2.

## Hahn

 18.23.1 ${{}_{1}F_{1}}\left({-x\atop\alpha+1};-z\right){{}_{1}F_{1}}\left({x-N\atop% \beta+1};z\right)=\sum_{n=0}^{N}\frac{{\left(-N\right)_{n}}}{{\left(\beta+1% \right)_{n}}n!}Q_{n}\left(x;\alpha,\beta,N\right)z^{n},$ $x=0,1,\dots,N$.
 18.23.2 ${{}_{2}F_{0}}\left({-x,-x+\beta+N+1\atop-};-z\right)\*{{}_{2}F_{0}}\left({x-N,% x+\alpha+1\atop-};z\right)=\sum_{n=0}^{N}\frac{{\left(-N\right)_{n}}{\left(% \alpha+1\right)_{n}}}{n!}Q_{n}\left(x;\alpha,\beta,N\right)z^{n},$ $x=0,1,\dots,N$.

## Krawtchouk

 18.23.3 $\left(1-\frac{1-p}{p}z\right)^{x}(1+z)^{N-x}=\sum_{n=0}^{N}\genfrac{(}{)}{0.0% pt}{}{N}{n}K_{n}\left(x;p,N\right)z^{n},$ $x=0,1,\dots,N$.

## Meixner

 18.23.4 $\left(1-\frac{z}{c}\right)^{x}(1-z)^{-x-\beta}=\sum_{n=0}^{\infty}\frac{{\left% (\beta\right)_{n}}}{n!}M_{n}\left(x;\beta,c\right)z^{n},$ $x=0,1,2,\dots$, $|z|<1$.

## Charlier

 18.23.5 ${\mathrm{e}}^{z}\left(1-\frac{z}{a}\right)^{x}=\sum_{n=0}^{\infty}\frac{C_{n}% \left(x;a\right)}{n!}z^{n},$ $x=0,1,2,\dots$.

## Continuous Hahn

 18.23.6 ${{}_{1}F_{1}}\left({a+\mathrm{i}x\atop 2\Re a};-\mathrm{i}z\right){{}_{1}F_{1}% }\left({\overline{b}-\mathrm{i}x\atop 2\Re b};\mathrm{i}z\right)=\sum_{n=0}^{% \infty}\frac{p_{n}\left(x;a,b,\overline{a},\overline{b}\right)}{{\left(2\Re a% \right)_{n}}{\left(2\Re b\right)_{n}}}z^{n}.$

## Meixner–Pollaczek

 18.23.7 $(1-{\mathrm{e}}^{\mathrm{i}\phi}z)^{-\lambda+\mathrm{i}x}(1-{\mathrm{e}}^{-% \mathrm{i}\phi}z)^{-\lambda-\mathrm{i}x}=\sum_{n=0}^{\infty}P^{(\lambda)}_{n}% \left(x;\phi\right)z^{n},$ $|z|<1$.