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!