Convergence of series with q-Pochhammer symbol

Question

I am working with the series $$\sum_{j=1}^{\infty}\rho \beta^{j-1} \prod_{n=0}^{j-2}(1+\rho \beta^n)$$ where the empty product is assumed as usual to be 1 and $\beta, \rho \in \mathbb R^+$, with $\beta <1$. I know that the product can be replaced by the q-Pochhamer symbol, obtaining: $$\sum_{j=1}^{\infty}\rho \beta^{j-1} \left(-\rho; \beta\right)_{j-1}$$ and that the series converges since $\beta<1$. Anyone can help me in finding the best way to approximate this quantity? any pointer to related work?

Thanks for your help!

Answer 1

Let use define $$ F(\rho,\beta) := \sum_{j=1}^{\infty}\rho \beta^{j-1} \left(-\rho; \beta\right)_{j-1}.$$ Then we have $$ F(x,q) = \sum_{n=1}^\infty x^n q^{n(n-1)/2}/(q;q)_n = -1 +\prod_{n=0}^\infty (1 + xq^n).$$ This is the generating function of OEIS sequence A008289 where $|q|<1$ for convergence.