Sum of reciprocal Pochhammer symbols of a fixed exponent
From specialfunctionswiki
Theorem
The following formula holds: $$\displaystyle\sum_{k=1}^n \dfrac{1}{(k)_p} = \dfrac{1}{(p-1)\Gamma(p)} - \dfrac{n\Gamma(n)}{(p-1)\Gamma(n+p)},$$ where $(k)_p$ denotes the Pochhammer symbol and $\Gamma$ denotes the gamma function.
