summation of Newton symbol product

Question

Anyone has idea how to simplify this summation: $$\sum\limits_{k=0}^{N}\binom{M-1+k}{k}\binom{M-1+N-k}{N-k}$$

Answer 1

Note that for $|x|<1$, and for any positive integer $m$, $$\frac{1}{(1-x)^{m}}=\sum_{i=0}^{\infty}\binom{-m}{i}(-x)^i =\sum_{k=0}^{\infty}\binom{m-1+k}{k}x^k.$$ Hence, by the Cauchy product of series, $$\sum\limits_{k=0}^{N}\binom{M-1+k}{k}\binom{M-1+N-k}{N-k}$$ is the coefficient of $x^N$ of the product $$\frac{1}{(1-x)^{M}}\cdot \frac{1}{(1-x)^{M}}=\frac{1}{(1-x)^{2M}}.$$ By the above formula, such coefficient is equal to $$\binom{2M-1+N}{N}.$$