# Mathematical Infinite Series - closed form

I have a question regarding the infinite series. I want to find a closed form expression for the following expression:

$$\sum_{n=1}^{+\infty}\frac{1}{\sqrt{n}}\frac{z^n}{n!}$$

where $z$ denotes the parameter. I don't know about the existence of any closed form expression or at least any approximation for the above series. I will be glad to see your comments and helps.

• Neither do I. Please let me know if you find one. – Lord Shark the Unknown Dec 2 '17 at 11:04
• – Gabriel Romon Dec 2 '17 at 11:08
• @GabrielRomon Thanks alot! I studied the comments of your problem. However it seems that there are some mistake on the proposed solutions or at least I am not convinced with some of them. I am wondering how we could converge to a solution! – Ali Dec 2 '17 at 13:30
• There are no simple closed forms, if not in terms of fractional Touchard polynomials. On the other hand the asymptotic behaviour for $x\to +\infty$ is pretty simple to study by squaring, see the question linked by Gabriel Romon. – Jack D'Aurizio Dec 2 '17 at 17:01
• If you follow the advice tired gives in a comment to the OP in the other thread to express $\frac 1{\sqrt n}=\frac 1{\sqrt \pi}\int_0^\infty e^{-nq^2}dq$, you get $$\frac 1{\sqrt \pi}\int_0^\infty e^{ze^{-q^2}}dq$$ – Paul Sinclair Dec 2 '17 at 17:05