Show $\sum_{n=0}^\infty \frac{(cq^n)_{\infty}}{(bq^n)_{\infty}}=\sum_{m=0}^\infty \frac{(c/b)_mb^mq^{nm}}{(q)_m}$
This was taken from the following equation: $$\frac{(b)_\infty}{(c)_\infty} \sum_{n=0}^\infty \frac{(a)_nt^n}{(q)_n} \cdot \frac{(cq^n)\infty}{(bq^n)\infty}=\frac{(b)_\infty}{(c)_\infty} \sum_{n=0}^\infty \sum_{m=0}^\infty \frac{(a)_nt^n}{(q)_n} \cdot \frac{(c/b)_mb^mq^{nm}}{(q)_m}$$
I have checked a closed resembling question for an answer, and I understand the first part based on the pochammer identity from reducing an infinite product to finite. However, like the question of this post, how does this relate to a new index $m$?Seeking an elegant proof that these two infinite series are equal