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How can I prove that: \begin{align} (ab;q)_n = \sum_{k=0}^{n} {n \choose k}_q (a;q)_{n-k} (b;q)_k \end{align} and \begin{align} \sum_{k\geq 0} u^k \cdot \frac{(a;q)_k}{(q;q)_k} = \frac{(u/q,1/q)_{\infty}}{(au/q ,1/q)_{\infty}} \end{align} I have tried hardly to solve that but fianlly i was lost and got nowhere. I'll appreciate any help.

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  • $\begingroup$ I've fixed the LaTeX, but what have you tried and what do you know about the $q$-binomial coefficients and $q$-series? (This is rather relevant; it is easy to solve problems like this using advanced material.) $\endgroup$ – darij grinberg Jan 14 at 10:46
  • $\begingroup$ I tried using things i know like simplifying the q- binomial and the (a,q) , (b,q) but i was stuck and did not succeed @darij grinberg $\endgroup$ – Almaa Jan 14 at 11:12
  • $\begingroup$ What formulas do you already know related to Pochhammer products and q-binomial coefficients? $\endgroup$ – darij grinberg Jan 15 at 0:01
  • $\begingroup$ Also, the first formula looks like the binomial formula, which can be proved by induction. Have you tried mimicking that induction proof? $\endgroup$ – darij grinberg Jan 15 at 0:04
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    $\begingroup$ I think you're right! The first identity cannot hold, at least with the definition of $q$-Pochhammer products I'm familiar with. (These are what I meant by Pochhammer products.) $\endgroup$ – darij grinberg Jan 15 at 7:53