Properties of Lerch Sums and Ramanujan's Mock Theta Functions

TL;DR

This paper investigates properties of Ramanujan's mock theta functions expressed as Lerch sums, demonstrating their integral representations, modular relations, and Fourier coefficient evaluations.

Contribution

It introduces new integral representations of Lerch sums as Jacobian theta functions and the sec-function, along with modular relations and Fourier coefficient evaluations.

Findings

Lerch sums can be expressed as integrals of Jacobian theta functions.

Established modular relations for Lerch sums.

Evaluated Fourier coefficients of certain Lerch sums.

Abstract

In this article we study properties of Ramanujan's mock theta functions that can be expressed in Lerch sums. We mainly show that each Lerch sum is actually the integral of a Jacobian theta function (here we show that for $\vartheta_3(t,q)$ and $\vartheta_4(t,q)$) and the $\sec-$function. We also prove some modular relations and evaluate the Fourier coefficients of a class of Lerch sums.