For integers $a$ and $b$, I am curious how to simplify an expression of the form $$\sum_{k=1}^n (-1)^k (a+k)! (b+k)!$$
I assume there is some simplification using properties of gamma and beta functions, as in this answer, but I am not well-versed enough in that type of number theory.
Any help is appreciated!
This is not an answer.
Factorials are growing so fast that, more than likely, the sum can be approximated by its last term, that is to say $$S_{a,b,n}=\sum_{k=1}^n (-1)^k (a+k)!\, (b+k)! \sim (-1)^n (a+n)!\, (b+n)!=T_{a,b,n}$$
Trying for $a=3$, $b=5$, a few values $$\left( \begin{array}{ccc} n & T_{a,b,n} & S_{a,b,n} \\ 1 & -17280 & -17280 \\ 2 & 604800 & 587520 \\ 3 & -29030400 & -28442880 \\ 4 & 1828915200 & 1800472320 \\ 5 & -146313216000 & -144512743680 \\ 6 & 14485008384000 & 14340495640320 \\ 7 & -1738201006080000 & -1723860510439680 \\ 8 & 248562743869440000 & 246838883359000320 \\ 9 & -41758540970065920000 & -41511702086706919680 \\ 10 & 8142915489162854400000 & 8101403787076147480320 \\ 11 & -1824013069572479385600000 & -1815911665785403238119680 \\ 12 & 465123332740982243328000000 & 463307421075196840089880320 \end{array} \right)$$