A note on two of Ramanujan’s mock theta functions

Abstract

In the “Lost” Notebook Ramanujan defined two mock theta functions (which we denote by \(\phi _{{}_R}\) and \(\xi _{{}_R}\)) and gave relations connecting them to some of the sixth-order mock theta functions. It turns out that these mock theta functions are related to each other by modular transformation. Their transformation formulas are similar to the transformation formulas for the second-order mock theta functions, and involve the same Mordell integrals. We prove several relations involving \(\phi _{{}_R}\) and \(\xi _{{}_R}\), and some relations connecting them to some of the second-order mock theta functions. Some alternate formulas for the second-order mock theta function \(\mu \) are given.