Here is a power series, which looks a bit like a Hypergeometric function series, but I don't think that it is. Has anyone any idea what it is? Here $n,p,r$ are integers with $n\ge 0$ and $p\ge r\ge 0$:


$$
f\_{n,p,r}(x)\,=\,\sum\_{s=0}^p \frac{x^s}{s!}\ \frac{p!\,(2n+p+s+2)!\,(n+r+s+2)!}{(p-s)!\,(2n+r+s+3)!\,(n+s+2)!}
$$


Originally this occurred as a $q$-factorial series, but if the $q=1$ case given here could be recognised, it would be a big help.


Thanks for the answer below. A bit before I declare this answered (if I remember how to do that!) I would like to sneak in another question. Is there any sensible way to sum this in the case $x=-1$? The answer is likely zero if $r<p$ and reasonably simple if $r=p$, but how to prove that?



Identifying a special function from its power series
Henry is almost right; the sum can be written as 
$$\frac { \left( 2n+p+2 \right) !\, \left( n+r+2 \right)!}
{ \left( 2n+r+3 \right) !\, \left( n+2 \right) !}
{}\_{3}F\_{2}\left({{2n+p+3,n+r+3,-p\,\,\,}\atop {2n+r+4,n+3}}\Bigm |-x\right)$$
 For $x=-1$ (no need to assume $r\le p$) the sum can be evaluated by Saalschütz's theorem (also called the Pfaff-Saalschütz theorem), giving
 $$ \left( -1 \right) ^{p}\frac { \left( 2n+p+2 \right) !\, \left( n+r
+2 \right) !\,r!\, \left( n+p \right) !}{ \left( 2+n+p \right) !\,
 \left( 3+2n+r+p \right) !\, \left( r-p \right) !\,n!}.
$$


Saalschütz's theorem is probably the most commonly used hypergeometric summation formula after Vandermonde's theorem (and it has a $q$-analogue). Zeilberger's algorithm can be used to prove any hypergeometric summation formula, but it won't tell you what it is. (Though Saalschütz's theorem can be found in the book $A=B$ in section 3.5, and section 3.6 has some information on how to identify hypergeometric summation formulas.)


