I would like to calculate the sum of the following convergent series:
$$\sum_{n=0}^\infty \frac{\Gamma(k+n+1)(-y)^n}{\Gamma(kN+N+c+n)n!(k+n)(k\alpha+n\alpha+2)}$$
where $k$ is a positive real number and $N$, $c$, and $\alpha$ are positive integers.
I know the fraction in the above equation without the Gamma functions converges and its sum is already calculated in https://stats.stackexchange.com/questions/445330/probability-distribution-function-expressed-in-terms-of-a-divergent-series?noredirect=1.
I have been looking for ways to find the sum of a convergent series but no use.
Best Answer
The ratio of consecutive terms is given by
$$\frac{a_{n+1}}{a_n}=\frac{(n+k)(n\alpha+k\alpha+2)}{(n+kN+c+1)(n+1)(n\alpha+(k+1)\alpha+2)}(-y)$$
Hence the series is given by a generalized hypergeometric function:
$$S=\frac{\Gamma(k+1)}{k(k\alpha+2)\Gamma(kN+c+1)}{}_2F_2\left(k,k+\frac2\alpha;kN+c+1,k+1+\frac2\alpha;-y\right)$$
I wouldn't suspect this reduces much further.