Bailey chains and generalized Lambert series. I: Four identities of Ramanujan
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.
scientific article; zbMATH DE number 118425