Closed form for particular power series $\sum^\infty_{k=0}\frac{a^kk^y}{(Tk+b)!}$.

Question

A (senior) acquaintance of mine was trying, unsuccesfully, to find a closed expressions for

$$\sum^\infty_{k=0}\frac{a^kk^y}{(Tk+b)!}$$

where $a\neq0$, and $y$, $b$ and $T$ are positive integers.

Except for the trivial special cases, is such a thing known? Any help or pointers toward the literature would be appreciated.

Answer 1

We have the Mittag-Leffler function,

$$E_{\alpha,\beta}(z)=\sum_{k=0}^\infty{z^k\over\Gamma(\alpha k+\beta)}$$

Or,

$$E_{T,b+1}(e^t)=\sum_{k=0}^\infty{e^{kt}\over(Tk+b)!}$$

And by repeated differentiation:

$${\partial^y\over\partial^yt}E_{T,b+1}(e^t)=\sum_{k=0}^\infty{e^{kt}k^y\over(Tk+b)!}$$

And finally:

$${\partial^y\over\partial^yt}E_{T,b+1}(e^t)\bigg|_{a=e^t}=\sum_{k=0}^\infty{a^kk^y\over(Tk+b)!}$$