It is a part of log function. please help me to find the sum of this infinite series $$1+e^{2x}+e^{6x}+e^{12x}+e^{20x}+\ldots$$ or $$\sum_{n=1}^{\infty} e^{n(n-1) x}$$
I got this problem while deriving specific heat of solids with an-harmonic quantum state of a particle. In simpler case the series will be $$1+e^{x}+e^{2x}+e^{3x}+e^{4x}+\ldots$$ Which can be summed by geometric progression formula. $$({1-e^{x}})^{-1}$$ Here in this problem I tried all the special series that I know to sum the series. So, I need few directions to tackle the problem.
Your series converges (for $\text{Re}(x) < 0$) to $$\frac{\vartheta_2(0, e^x)}{2 e^{x/4}}$$ (in Abramowitz and Stegun's notation, also used by Maple), where $\vartheta_2$ is a Jacobi Theta function.