How do we represent the summation in the form of elementary functions??
$$ \sum_{n=0}^{\infty}{\frac{x^n \Gamma(n+a)}{n! \Gamma(n+b)}} $$
2026-04-07 00:20:14.1775521214
Series involving gamma functions
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From the definition: $$\Gamma(n+a)=(a)_n\Gamma(a)$$ where $(a)_n$ is the Pochhammer symbol.
So $$$$ $$\sum_{n=0}^{\infty}{\frac{x^n \Gamma(n+a)}{n! \Gamma(n+b)}}=\frac{\Gamma(a)}{\Gamma(b)}\sum_{n=0}^{\infty}{\frac{(a)_nx^n }{(b)_n n!}}=\frac{\Gamma (a) \,}{\Gamma (b)}\, _1F_1(a;b;x)$$ where $_1F_1(a;b;x)$ is confluent hypergeometric function.