Factorial of Sum of Natural Numbers

Question

Given natural numbers $x,y$, are there some identities between to $(x+y)!$, $x!$, $y!$, and some sum of "mixed" terms with $x$ and $y$? Essentially, is there a nice expansion of the terms if one were to expand,

$(x+y)!=(x+y)(x+y-1)\,\cdots\,(x+y-k)\,\cdots\,2\cdot1$

I can't seem to find anything relevant. I am open to generalizations to the gamma function if such identities exist in this more general context.

Answer 1

Some possible developments are $$ \begin{gathered} (n + m)!\quad \left| {\,n,m \in \,\mathbb{N}\,\;} \right. = \left( {n + m} \right)^{\,\underline {\,n + m\,} } = \hfill \\ = \left( {n + m} \right)^{\,\underline {\,n\,} } m^{\,\underline {\,m\,} } = \left( {m + 1} \right)^{\,\overline {\,n\,} } m^{\,\underline {\,m\,} } = \hfill \\ = m!\sum\limits_{0\, \leqslant \,k\, \leqslant \,\min \left( {n,m} \right)} {\left( \begin{gathered} n \\ k \\ \end{gathered} \right)\;n^{\,\underline {\,n - k\,} } \;m^{\,\underline {\,k\,} } } = \hfill \\ = m!\,n!\sum\limits_{0\, \leqslant \,k\, \leqslant \,\min \left( {n,m} \right)} {\;\frac{{n^{\,\underline {\,n - k\,} } \;m^{\,\underline {\,k\,} } }} {{\left( {n - k} \right)!k!}}} = \hfill \\ = m!\,n!\sum\limits_{0\, \leqslant \,k\, \leqslant \,\min \left( {n,m} \right)} {\;\left( \begin{gathered} n \\ k \\ \end{gathered} \right)\left( \begin{gathered} m \\ k \\ \end{gathered} \right)} = \hfill \\ = m!\,n!\left( \begin{gathered} n + m \\ m \\ \end{gathered} \right) \hfill \\ \end{gathered} $$ with $$ \begin{gathered} n^{\,\underline {\,q\,} } = n\left( {n - 1} \right) \cdots \left( {n - q + 1} \right):\text{falling}\,\text{factorial} \hfill \\ n^{\,\overline {\,q\,} } = n\left( {n + 1} \right) \cdots \left( {n + q - 1} \right):\text{rising}\,\text{factorial} \hfill \\ \end{gathered} $$

Answer 2

That's pretty broad... the only naturally useful fact I can think of is that $x!y!\mid (x+y)!$, and their quotient is the binomial coefficient $\binom{x+y}{x}=\binom{x+y}{y}$.