Let $\lambda, \mu \in \left (0, \frac {1} {2} \right ).$ Then determine whether the following sum converges or not $:$$$\sum\limits_{n = 0}^{\infty} \left \lvert \sum\limits_{m = 0}^{n} \frac {\Gamma (\lambda + m)} {m!} \frac {\Gamma (\lambda + n - m)} {(n - m)!} \right \rvert^2 \frac {\Gamma (n + 1)} {\Gamma (n + 2 - 2 \mu)}.$$
I found this series while working with an estimation of an integral involving Bergman kernel of the symmetrized bidisc (which I prefer not to disclose due to privacy constraint). I am trying to investigate whether the sum is finite failing to which the integral estimate I am looking for would not hold. But I have no idea how to make sure whether the sum is finite or not. I have tried to use Stirling's approximation but unfortunately that would lead me to nowhere.
Any help in this regard would be greatly appreciated. Thanks for your valuable time.