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I need references for

$\sum_{n=0}^N\frac{q^n}{(q^2;q^2)_n(q^2;q^2)_{N-n}}=\frac{(-q,q)_N}{(q^2;q^2)_N}$

and

$\sum_{n=0}^N\frac{(-1)^nq^{n^2}}{(q^2;q^2)_n(q;q)_{N-n}}=\frac1{(q^2;q^2)_N}$

A similar, but trivial identity is

$\sum_{n=0}^N\frac{(-1)^nq^{\binom{n+1}2}}{(q;q)_n(q;q)_{N-n}}=1$

which follows directly from the q-binomial theorem. Given the rough similarity of the left-hand-sides of these identities, I also wonder if these are part of a larger class of identities.

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I just received good references from Johann Cigler: 1st identity: J Cigler, Séminaire Lotharingien de Combinatoire, B05a (1981), BA Kupershmidt, Journal of Nonlinear Mathematical Physics 7, 244–262 (2000) with references to papers by Fine and Andrews 2nd identity: special case of Cauchy identity, WP Johnson, AM Monthly 111, 791‐800 (2004), with reference to Jacobi I still wonder whether these identities have a unified treatment or whether they are "singular". –  Thomas Prellberg Dec 9 '12 at 15:59

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