The Mordell integral, quantum modular forms, and mock Jacobi forms

摘要

Abstract It is explained how the Mordell integral ∫ R e πiτ x 2 ? 2 πzx cosh ( πx ) dx $$int_{mathbb R} rac{e^{pi i au x^{2} - 2pi zx}}{cosh(pi x)} dx $$ unifies the mock theta functions, partial (or false) theta functions, and some of Zagier’s quantum modular forms. As an application, we exploit the connections between q -hypergeometric series and mock and partial theta functions to obtain finite evaluations of the Mordell integral for rational choices of τ and z . Mathematics Subject Classification: 11P55, 05A17