Question: Is there a closed-form expression for the following sum

$$ F(z,k,r)=\sum_{n=0}^{r} \frac{z^n}{{n+k} \choose {k}}\label{sum}\tag{1} $$

where $z\in\mathbb{C}$, and $r$, $k$ are non-negative integers.

Remark: Obviously, the above expression is a polynomial of degree $r$. However, I am interested in "alternative" expressions for it as later on I will need to compute residues of the product of above functions (see below).

Motivation: I am trying to compute a much nastier expression involving sum over multisets and ratios of multinomial coefficients:

$$ \sum_{(n_1,\ldots,n_d)\vdash N-k}\dfrac{{{k} \choose {k_1,k_2,\ldots,k_d}} {{N-k} \choose {n_1,n_2,\ldots,n_d}} }{ {{N} \choose {n_1+k_1,n_2+k_2,\ldots,n_d+k_d}}}\label{ssum}\tag{2} $$

where $N,k$ are fixed non-negative integers and $(k_1,\ldots,k_d)$ is fixed and satisfies $(k_1,\ldots,k_d)\vdash k$.

I got expression $\eqref{sum}$ by trying to compute $\eqref{ssum}$ explicitly via the integral representation of Kronecker delta.

2more comments