# §18.23 Hahn Class: Generating Functions

For the definition of generalized hypergeometric functions see §16.2.

## Hahn

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

## Krawtchouk

 18.23.3 $\left(1-\frac{1-p}{p}z\right)^{x}(1+z)^{N-x}=\sum_{n=0}^{N}\binom{N}{n}\mathop% {K_{n}\/}\nolimits\!\left(x;p,N\right)z^{n},$ $x=0,1,\dots,N$.

## Meixner

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

## Charlier

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

## Continuous Hahn

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

## Meixner–Pollaczek

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