New proofs of some Dedekind η-function identities of level 6

Abstract

Recently, Shaun Cooper proved several interesting ?-function identities of level 6 while finding series and iterations for 1/?. In this sequel, we present some new proofs of the ?-function identities of level 6 discovered by Cooper. Here, in this article, we make use of the modular equation of degree 3 in two methods. We further give some interesting combinatorial interpretations of colored partitions. We also briefly describe a potential direction for further researches based upon some related recent developments involving the Jacobi?s triple-product identity and the theta-function identities as well as on several other q-functions which emerged from the Rogers-Ramanujan continued fraction R(q) and its such associates as G(q) and H(q). We point out the importance of the usage of the classical q-analysis and we also expose the current trend of falsely-claimed ?generalization? by means of its trivial and inconsequential (p, q)-variation by inserting a forced-in redundant (or superfluous) parameter p.