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I stumbled upon the following simple $q-$identities: $$\frac{1}{(-q;q)_\infty}\sum \limits_{j =0}^{\infty}\frac{q^{(2r+1)j}}{(q^2;q^2)_j}=(q;q^2)_r$$ and $$\frac{1}{(q;q^2)_\infty}\sum \limits_{j =0}^{\infty}(-1)^{j}\frac{q^{(r+1)j}}{(q;q)_j}=(-q;q)_r,$$ where $r\in\mathbb{N} $ and $\binom{r}{j}_q$ denotes a $q-$binomial coefficient.

Probably these are well known. But where can I find such identities? Are there tables of $q-$identities in the literature?

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You can use eqn. (1), (2) and (5) in this file to show the identities.

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  • $\begingroup$ Thanks for the advertisement! Yes, indeed, all those identities are particular cases of the $q$-binomial theorem... $\endgroup$ – Wadim Zudilin Nov 16 '15 at 11:58
  • $\begingroup$ @Satoshi: Thank you. I used essentially the same methods to prove these identities. But I want to know if the identities in the given form appear in the literature and moreover if somewhere are tables of known $q-$identities where possibly analogous formulae can be found. $\endgroup$ – Johann Cigler Nov 16 '15 at 12:01
  • $\begingroup$ These are simplest variants of Rogers-Ramanujan (Andrews-Gordon) identities and they are also called quantum dilogarithm identities. You can google these words to reach the original references. $\endgroup$ – Satoshi Nawata Nov 16 '15 at 16:06

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