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A new proof of some identities of Bressoud. (English) Zbl 1004.05008

Summary: We provide a new proof of the following two identities due to Bressoud: \[ \sum_{m=0}^N q^{m^{2}} \left[\begin{matrix} N\\m \end{matrix}\right] = \sum_{m=-\infty}^{\infty} (-1)^m q^{m(5m+1)/2} \left[\begin{matrix} 2N\\N+2m \end{matrix}\right], \]
\[ \sum_{m=0}^N q^{m^{2}+m} \left[\begin{matrix} N\\m \end{matrix}\right] = (1/ (1-q^{N+1})) \sum_{m=-\infty}^{\infty} (-1)^m\times q^{m(5m+3)/2} \left[\begin{matrix} 2N+2\\N+2m+2 \end{matrix}\right], \] which can be considered as finite versions of the Rogers-Ramanujan identities.

MSC:

05A19 Combinatorial identities, bijective combinatorics
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