Let $m, n\in \mathbb{N}$ and $|x| < 1$. I look for hints to derive an analytic formula for
$$f\_{m,n}(x) = \sum\_{k \in \mathbb{N}} {n + k \choose k} {m + k \choose k} x^{k}. $$



Generating function of product of binomial coefficients
This is Gauss' hypergeometric function
$F(n+1,m+1,1;x)$. You can then apply the huge theory of hypergeometric functions to derive further expressions. For instance, Euler's transformation formula gives the alternative expression
$$\frac 1{(1-x)^{m+n+1}}\,F(-m,-n,1;x)=\frac 1{(1-x)^{m+n+1}}\sum\_{k=0}^{\min(m,n)}\binom mk\binom nk x^k$$
for the same series as a finite sum.


