Prove that $\sum_{-\infty}^{\infty}\frac{(a;q)_nt^n}{(b;q)_n}$ is an analytic funciton in $b$.

Question

In the proof of the theorem $$\sum_{-\infty}^{\infty}\frac{(a;q)_nt^n}{(b;q)_n}=\frac{(at,\frac{q}{at},q,\frac{b}{a};q)_\infty}{(t,\frac{b}{at},b,\frac{q}{t};q)_\infty}$$ It is used both sides are analytic in $b$. When $|b/a|<|t|<1$ and $|q|<1$. I can see the right hand side is analytic, but i do not see how the left hand side is. How would i prove this? I assume I need to use the Wierstrass M-test, but do not see a nice bound here. Any help would be appreciated.

Answer 1

Each term is analytic and it converges locally uniformly (Morera's theorem).