Binomial basis of polynomial ring

Question

The polynomial ring $\mathbb{Q}[x]$ has a basis given by the "binomial powers" $\binom{x}{0}, \binom{x}{1}, \binom{x}{2}, \ldots$ where we define $$ \binom{x}{k} = \frac{1}{k!} x (x - 1) \ldots (x - k + 1).$$ Given $n, m \geq 0$, is there a nice expression for the product of two of these basis elements $\binom{x}{n} \binom{x}{m}$ in terms of the binomial basis?