Recently I gave Mathematica the following input on the left hand side
$$ \sum_{n=0}^{\infty}\frac{\Gamma(a+n)}{\Gamma(b+n)}=\frac{\Gamma(a)\Gamma(b-a-1)}{\Gamma(b-1)\Gamma(b-a)}. $$
Can anyone explain to me what identities are needed to get the expression on the right hand side above?