# Reference request for a certain exponential series

I recently encountered the series $$\sum_{d \in \mathbb{Z}} e^{-t^d}t^{kd},$$ for real $$0 and $$k$$ a positive integer. It converges because roughly speaking, when $$d$$ is large and positive the terms are about $$t^{kd}$$, and when $$d$$ is large and negative $$e^{-t^d}$$ is very small. Has it or a series like it been studied in the literature to anyone’s knowledge, and are there reasonable alternative representations of it?

## 1 Answer

Q: Is there a reasonable alternative expression for the sum $$\sum_{d=-\infty}^\infty e^{-t^d}t^{kd},\;\;0 A: I note the integral expression $$\int_{-\infty}^\infty e^{-t^x}t^{kx}\,dx=-\frac{(k-1)!}{\ln t}.$$ This is a reasonable approximation to the sum when $$t$$ is not much smaller than 1 and $$k$$ is a small integer. For $$k=1$$, in particular, sum and integral are nearly indistinguishable for $$t\gtrsim 0.2$$. If $$t\gtrsim 0.5$$ one can go up to $$k=8$$.

Sum (gold) versus integral (blue) as a function of $$t$$, for $$k=1,2,4,8$$. The oscillations should follow from the first few terms of the Poisson summation (the integral being the zeroth term). I have not succeeded in evaluating these.