Separation in the q-shifted factorial

Question

Is their any formula to separate the power in $$(q^{x+y};q)_n$$

where $(a;q)_n=\prod_{i=0}^{n-1}(1-aq^i)$ denotes the $q$-shifted factorial.

P.S: $x,y$ and $n$ are all integers.

I'm glade for your help.

Answer 1

You can write this as \begin{align} (q^{x + y}; q)_n &= \prod_{i = 0}^{n - 1} (1 - q^{x + y}q^i) \\ &= \prod_{j = x}^{n + x - 1} (1 - q^yq^j) \qquad (\text{where }j = i + x) \\ &= \frac{\prod_{j = 0}^{n + x - 1} (1 - q^yq^j)}{\prod_{j = 0}^{x - 1} (1 - q^yq^j)} \\ &= \frac{(q^y;q)_{n + x}}{(q^y;q)_x}. \end{align}

Another option is $$ (q^{x + y};q)_n = \frac{(q;q)_{n+x+y-1}}{(q;q)_{x+y-1}}. $$

But I don't think you can separate $x$ and $y$ any more so that you can separate them in $(x + y)!$. After all,

$$ (q;q)_n = [n]_q!(1 - q)^{n} $$

so what you have is

$$ (q^{x + y};q)_n = \frac{[n + x + y -1]_q!}{[x + y - 1]_q!}(1-q)^n.$$