A fundamental modular identity and some applications

Abstract:

We prove a six-parameter identity whose terms have the form ${x^\alpha }T({k_1},{l_1})T({k_2},{l_2})$, where $T(k,l) = \sum \nolimits _{ - \infty }^\infty {{x^{k{n^2} + l n}}}$. This identity is then used to give a new proof of the familiar Ramanujan identity $H(x)G({x^{11}}) - {x^2}G(x)H({x^{11}}) = 1$, where $G(x) = \prod \nolimits _{n = 0}^\infty {{{[(1 - {x^{5n + 1}})(1 - {x^{5n + 4}})]}^{ - 1}}}$ and $H(x) = \prod \nolimits _{n = 0}^\infty {{{[(1 - {x^{5n + 2}})(1 - {x^{5n + 3}})]}^{ - 1}}}$. Two other identities, called "balanced ${Q^2}$ identities", are also established through its use.