What does it mean to say 'removing the first factor ' from the following function :
$$f(a)=\int_0^{\infty}t^{x-1}\frac{(-at;q)_{\infty}dt}{(-t;q)_{\infty}}$$
This interval converges when $x>0$ and $|a|<q^x$. Assume at the moment that $x\ne1, 2 ...$.
By "removing the first factor" of $(-at;q)_{\infty}$, I have $$f(a)=\int_0^{\infty}t^{x-1}\frac{(-aqt;q)_{\infty}}{(-t;q)_{\infty}}[1+a(t+1)-a]dt$$
So I have checked the pochammer equivalent, and I have
$$\left(-at;q\right)=\prod_{k=0}^{\infty}(1+atq^k)=(1+at)(1+atq)(1+\cdots)$$
Now If I remove $(1+at)$, then am I to show the following are equal to? $$(1+atq)(1+atq^2)(1+\cdots)=(-aqt;q)_{\infty}[1+a(t+1)-a]$$
Ok so I somewhat answered my own question, we have by the following $$\prod_{k=0}^{\infty}\frac{(1+atq^k)}{(1+at)}=\prod_{k=1}^{\infty}(1+atq^k) =\prod_{k=1}^{\infty}(1+atq^k)[1+a(t+a)-a]=(-aqt;q)_{\infty}[1+a(t+1)-a]$$