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On the Andrews-Schur proof of the Rogers-Ramanujan identities. (English) Zbl 1218.05010

Summary: In this article, we study one of Andrews’ proofs of the Rogers-Ramanujan identities published in 1970 [G.E. Andrews, “A polynomial identity which implies the Rogers-Ramanujan identities,” Scripta Math. 28, 297–305 (1970; Zbl 0204.32403)]. His proof inspires connections to some famous formulas discovered by Ramanujan. During the course of study, we discovered identities such as \[ \sum_{n\geq0}\frac{q^{n^2}}{(q;q)_n}=\frac{1}{\sqrt{5}}\Biggl(\beta \prod_{n=1}^{\infty}\frac{1}{1+\alpha q^{n/5}+q^{2n/5}}-\alpha \prod_{n=1}^{\infty}\frac{1}{1+\beta q^{n/5}+q^{2n/5}}\Biggr), \] where \(\beta = - 1 / \alpha \) is the Golden Ratio.

MSC:

05A15 Exact enumeration problems, generating functions
05A30 \(q\)-calculus and related topics
05A40 Umbral calculus

Citations:

Zbl 0204.32403

Software:

recpf; RRtools
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Full Text: DOI

References:

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