Factorial of Sum of Natural Numbers 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.
 A: 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 $\binom{x+y}{x}=\binom{x+y}{y}$.
A: 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
$$
\begin{gathered}
  n^{\,\underline {\,q\,} }  = n\left( {n - 1} \right) \cdots \left( {n - q + 1} \right):\text{falling}\,\text{factorial} \hfill \\
  n^{\,\overline {\,q\,} }  = n\left( {n + 1} \right) \cdots \left( {n + q - 1} \right):\text{rising}\,\text{factorial} \hfill \\ 
\end{gathered} 
$$
