Modular transformations involving the Mordell integral in Ramanujan's lost notebook

Youn-Seo Choi

Vol. 272 (2014), No. 1, 59–85

DOI: 10.2140/pjm.2014.272.59

Abstract

In his “lost notebook” (p. 202 of the 1988 edition), S. Ramanujan recorded modular transformations involving the Mordell integral, $q$-hypergeometric series, and generalized Lambert series. He gave no proofs; here we prove these formulas and use them to derive modular transformations of third-order mock theta functions. Mordell’s formula, the properties of $q$-hypergeometric series and Appell–Lerch sums play central roles in the proofs.

For my teacher Bruce C. Berndt on his 75th birthday.

Keywords

Mordell integral, Appell–Lerch sum, Ramanujan

Mathematical Subject Classification 2010

Primary: 11B65

Secondary: 11F27, 33D99

Milestones

Received: 9 April 2013

Revised: 19 August 2013

Accepted: 23 September 2013

Published: 9 October 2014