Series $\sum_{n=0}^\infty \frac{a^n}{\Gamma(b+nc)}$

Question

$a,b,c$ be positive real numbers, and $\Gamma$ is the Gamma function, find

$$ \sum_{n=0}^\infty \frac{a^n}{\Gamma(b+nc)} $$

Answer 1

Providing solution in the $c=1$ case. Let me know in the comments if $a,b,c$ are natural numbers, in that case it can be solved without any such assumption, but even there the underlying idea won't change. $$ \sum_{n=0}^\infty \frac{a^n}{\Gamma(b+n)} $$ Now as $$\Gamma(x+1)=x\Gamma(x)$$ This gives us on repeated application $$\Gamma(b+n)=\prod_{k=0}^{n-1 }{(b+k)} \Gamma(b)$$ Therefore our sum becomes $$\frac{1}{\Gamma(b)} \sum_{n=0}^{\infty}{\frac{a^n}{\prod_{k=0}^{n-1 }{(b+k)}}}$$

Can you take it from here; does the sum look familiar now?