A finite summation of double binomial coefficients I find the following identity and have checked on Mathematica, while I have no idea how to prove it:
$$\sum_{j=0}^n(-1)^{n-j}\binom{p+j}{j}\binom{n+\beta}{n-j}=\binom{p-\beta}{n}, \quad \beta>-1, \quad p>\beta-1.$$
It seems that proof by induction does not work and because of the gamma function, the transform technique does not work either.

We have
$$
\eqalign{
  & \sum\limits_{j = 0}^n {\left( { - 1} \right)^{\,n - j} \left( \matrix{
  p + j \cr 
  j \cr}  \right)\left( \matrix{
  n + \beta  \cr 
  n - j \cr}  \right)}  =   \cr 
  &  = \sum\limits_{j = 0}^n {\left( { - 1} \right)^{\,n} \left( \matrix{
   - p - 1 \cr 
  j \cr}  \right)\left( \matrix{
  n + \beta  \cr 
  n - j \cr}  \right)}  =   \cr 
  &  = \left( { - 1} \right)^{\,n} \sum\limits_{\left( {0\, \le } \right)\,j\,\left( { \le \,n} \right)} {\left( \matrix{
   - p - 1 \cr 
  j \cr}  \right)\left( \matrix{
  n + \beta  \cr 
  n - j \cr}  \right)}  =   \cr 
  &  = \left( { - 1} \right)^{\,n} \left( \matrix{
   - p - 1 + n + \beta  \cr 
  n \cr}  \right) =   \cr 
  &  =  \left( \matrix{
  p - \beta  \cr 
  n \cr}  \right)\quad \left| \matrix{
  \;n \in Z \hfill \cr 
  \;\forall p,\beta  \hfill \cr}  \right. \cr} 
$$
where the steps are:
 - upper negation (always valid for integer $j$);
 - we can omit summming limits, because they are implicit in the two binomials;
 - convolution;
 - upper negation (always valid for integer $n$).   
