Question Number 128321 by Dwaipayan Shikari last updated on 06/Jan/21
$${Prove} \\ $$$$\underset{{n}\geqslant\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left({a}\right)_{{n}} \left({b}\right)_{{n}} }{\left({c}\right)_{{n}} {n}!}=\frac{\Gamma\left({c}\right)\Gamma\left({c}−{a}−{b}\right)}{\Gamma\left({c}−{a}\right)\Gamma\left({c}−{b}\right)} \\ $$$${Where}\:\left({a}\right)_{{n}} =\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\left({k}+{a}\right) \\ $$
Commented by Lordose last updated on 10/Jan/21
$$\mathrm{hypergeometric}\:\mathrm{series} \\ $$