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I'm afraid I can't deduce a simple expresion for the following infite serie $\sum_{i=0}^{\infty} p^{i(i-1)/2} r^i,$ with $p,r <1$.

Since $p^{i^2} < p^i < 1$, I think the serie converges. I've tried to use the tricks for geometric and other well-known series, but I'm afraid they are useless. Also I've consult books like Hirschman and Hyslop, but I haven't found anything.

This series is the renormalization constant for the stationary distribution of a birth-death chain, with $\lambda_i=\lambda p^i, \mu_i=\mu,$ with $\lambda, \mu$ positive constants, which are the birth and death rates respectively, provided $r=(\lambda / \mu) <1.$

I would appreciate any suggestions, have a nice day!

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I don't think there is a simple expression. In the special case $r=1$ you're looking at $\sum p^{i(i-1)/2}$ which is an example of what is known as a theta function. There is a lot of literature on theta functions, but they can't be reduced to exponentials, logarithms, trig functions, etc.