Polynomial defined by products of binomial coefficients

Question

Consider the polynomial in $x$, $$\sum_{k=1}^n {n \choose k} {m+k-1 \choose m} x^k , $$ where $m$ and $n$ are positive non-zero integers.

Question: can it be expressed in terms of known function(s) of $x$?

Answer 1

If you want the generic function$$f_{m,n}(x)=\sum_{k=1}^n {n \choose k} {m+k-1 \choose m} x^k$$ $$f_{m,n}(x)=n\, x \,\, _2F_1(m+1,1-n;2;-x)$$ where appears the Gaussian or ordinary hypergeometric function (see here).

A few expressions $$\left( \begin{array}{cc} m & f_{m,n}(x) \\ 0 & (x+1)^n-1 \\ 1 & n x (x+1)^{n-1} \\ 2 & \frac{1}{2} n x (x+1)^{n-2} ((n+1) x+2) \\ 3 & \frac{1}{6} n x (x+1)^{n-3} \left((n^2 +3 n +2) x^2+6 (n+1) x+6\right) \end{array} \right)$$ which are just polynomials of degree $n$.

If you define $$g_{m,n}(x)=\frac{m!} {n \,x\, (x+1)^{n-m}}\, f_{m,n}(x)=P_{m-1}(x)$$