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

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