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Let us define this sum as a function of $z \in \mathbb{C}$ with some positive parameter $a$ $$ f(z; a) = \sum\limits_{n = 0}^{\infty}\frac{|z|^{2n}}{n!}e^{-ian^2}. $$ Probably, it can be expressed (or ...

Let $0<a\neq 1$ be a fixed real number and denote by $a^{ \frac{x}{}}:=\mbox{uxp}_a(x)$ the ultra exponential function (ultra power) that is a unique extension of tetration (and also its linear ...

I already asked this question here https://math.stackexchange.com/questions/2005211/estimate-of-sum-n-1-infty-exp-pi-i-t-n/2005778#2005778 but I did not get a satisfying answer, so I put it here. I ...