Consider the sum
$F(a_1,\cdots,a_r; z)\sum_{m=0}^{\infty}\frac{(a_1+m)(a_2+m)\cdots(a_r+m)}{(a_1)(a_2)\cdots (a_r)}\frac{z^m}{m!},$
where $a_i$ for $i=1,2,\cdots,r$ is an increasing sequence of positive integers. And yes the denominator is independendent of $m$. In some cases the ratio becomes hypergeometric when restricting the sum over a particular set of indices. Surely such functions have been studied before? Do they have a common name? Is there a nice integral representation?