Bailey chains and generalized Lambert series. I: Four identities of Ramanujan
type: Item
instance of: scholarly article
title: Bailey chains and generalized Lambert series. I: Four identities of Ramanujan
zbMATH Open document ID: 0773.05007
author: George E. Andrews
published in: Illinois Journal of Mathematics
publication date: 1993-02-10T00:00:00Z
review text: The author examines four identities of Ramanujan like \[ {1\over\varphi^ 2(-q)}\sum^ \infty_{n=1}(-1)^{n-1}n^ 2q^{{n+1\choose 2}}=\sum^ \infty_{n=1}{q^{n(2n-1)}(1+q^{2n-1})\over(1-q^{2n- 1})^ 2} \] and \[ {1\over\psi^ 2(q)}\sum^ \infty_{n=1}{(-1)^{n- 1}(2n-1)^ 2q^{n(n-1)}(1+q^{2n-1}) \over 1-q^{2n-1}}=1+8\sum^ \infty_{n=1}{(-1)^ nq^{n(n+1)}\over(1+q^ n)^ 2}; \] where \[ \varphi(-q)=\sum^ \infty_{n=-\infty}(-1)^ nq^{n^ 2}=\sum^ \infty_{n=1}{(1-q^ n) \over(1+q^ n)}, \] and \[ \psi(q)=\sum^ \infty_{n=0}q^{{n+1\choose 2}}=\prod^ \infty_{n=1}{(1- q^{2n})\over(1+q^{2n-1})}. \] These identities appear to be closely related to the first five identities and other results of Ramanujan in his note books, they seem to be much deeper requiring intricate proofs. The key elements of their proofs are (1) series rearrangement, (2) Bailey pairs, (3) \(q\)-series transformations. These identities closely resemble formulas of G. Humbert for generating functions \({\mathcal A}(q)\) and \({\mathcal B}(q)\) of class numbers related to binary quadratic forms.
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 11P99
zbMATH DE Number: 118425
zbMATH Keywords: Bailey chains
zbMATH Keywords: generalized Lambert series
zbMATH Keywords: Ramanujan identities
zbMATH Keywords: series rearrangement
zbMATH Keywords: Bailey pairs
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 118425