A reciprocity theorem for certain q-series found in Ramanujan's lost notebook

Bruce C. Berndt; Song Heng Chan; Boon Pin Yeap; Ae Ja Yee

doi:10.1007/s11139-006-0241-5

A reciprocity theorem for certain q-series found in Ramanujan's lost notebook

Bruce C. Berndt, Song Heng Chan, Boon Pin Yeap, Ae Ja Yee

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Three proofs are given for a reciprocity theorem for a certain q-series found in Ramanujan's lost notebook. The first proof uses Ramanujan's ₁ψ₁ summation theorem, the second employs an identity of N. J. Fine, and the third is combinatorial. Next, we show that the reciprocity theorem leads to a two variable generalization of the quintuple product identity. The paper concludes with an application to sums of three squares.

Original language	English (US)
Pages (from-to)	27-37
Number of pages	11
Journal	Ramanujan Journal
Volume	13
Issue number	1-3
DOIs	https://doi.org/10.1007/s11139-006-0241-5
State	Published - Jun 2007

Keywords

Combinatorial bijection
Quintuple product identity
Ramanujan's ψ summation theorem
Reciprocity theorem
Sums of three squares
q-series

ASJC Scopus subject areas

Algebra and Number Theory

Online availability

10.1007/s11139-006-0241-5

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title = "A reciprocity theorem for certain q-series found in Ramanujan's lost notebook",

abstract = "Three proofs are given for a reciprocity theorem for a certain q-series found in Ramanujan's lost notebook. The first proof uses Ramanujan's 1ψ1 summation theorem, the second employs an identity of N. J. Fine, and the third is combinatorial. Next, we show that the reciprocity theorem leads to a two variable generalization of the quintuple product identity. The paper concludes with an application to sums of three squares.",

keywords = "Combinatorial bijection, Quintuple product identity, Ramanujan's ψ summation theorem, Reciprocity theorem, Sums of three squares, q-series",

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note = "Funding Information: A. J. Yee: Research partially supported by a grant from The Number Theory Foundation. Funding Information: B. C. Berndt: Research partially supported by grant MDA904-00-1-0015 from the National Security Agency.",

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N2 - Three proofs are given for a reciprocity theorem for a certain q-series found in Ramanujan's lost notebook. The first proof uses Ramanujan's 1ψ1 summation theorem, the second employs an identity of N. J. Fine, and the third is combinatorial. Next, we show that the reciprocity theorem leads to a two variable generalization of the quintuple product identity. The paper concludes with an application to sums of three squares.

AB - Three proofs are given for a reciprocity theorem for a certain q-series found in Ramanujan's lost notebook. The first proof uses Ramanujan's 1ψ1 summation theorem, the second employs an identity of N. J. Fine, and the third is combinatorial. Next, we show that the reciprocity theorem leads to a two variable generalization of the quintuple product identity. The paper concludes with an application to sums of three squares.

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A reciprocity theorem for certain q-series found in Ramanujan's lost notebook

Abstract

Keywords

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