Table 1:

Examples of GHPDs and their probability generating functions

Distributions	Probability generating function	Parameters
Binomial	$\frac{{{}_{1}F}_0(-n;;-ps/(1-p\left)\right)}{{{}_{1}F}_0(-n;;-p/(1-p\left)\right)}$	$0<p<1$
Poisson	$\frac{{{}_{0}F}_0(;;\theta s)}{{{}_{0}F}_0(;;\theta )}$	$\theta >0$
Hyper-Poisson	$\frac{{{}_{1}F}_1(1;\psi ;\theta s)}{{{}_{1}F}_1(1;\psi ;\theta )}$	$\psi >0,\theta >0$
Geometric	$\frac{{{}_{1}F}_0(1;;qs)}{{{}_{1}F}_0(1;;q)}$	$0<q<1$
Negative Binomial	$\frac{{{}_{1}F}_0(k;;qs)}{{{}_{1}F}_0(k;;q)}$	$k>\mathrm{0,\; 0}<q<1$
Hypergeometric	$\frac{{{}_{2}F}_1(-n,-Np;N-Np-n+1;s)}{{{}_{2}F}_1(-n,-Np;N-Np-n+1;1)}$	$n,N\in \mathbb{N},0<p<1$
Beta-Negative Binomial	$\frac{{{}_{2}F}_1(k,\ell ;k+\ell +m;s)}{{{}_{2}F}_1(k,\ell ;k+\ell +m;1)}$	$k,\ell ,m\ge 0$
Extended Generalized Waring	$\frac{{{}_{2}F}_1(k,\ell ;k+\ell +m;\theta s)}{{{}_{2}F}_1(k,\ell ;k+\ell +m;\theta )}$	$k,\ell >0,m\in \mathbb{R},0<\theta <1$