The Mordell Integral, Quantum Modular Forms, and Mock Jacobi Forms
نویسندگان
چکیده
It is explained how the Mordell integral ∫ R e −2πzx cosh(πx) 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. 1. The Mordell Integral The Mordell integral is
منابع مشابه
Unimodal Sequences and “strange” Functions: a Family of Quantum Modular Forms
In this paper, we construct an infinite family of quantum modular forms from combinatorial rank “moment” generating functions for strongly unimodal sequences. The first member of this family is Kontsevich’s “strange” function studied by Zagier. These results rely upon the theory of mock Jacobi forms. As a corollary, we exploit the quantum and mock modular properties of these combinatorial funct...
متن کاملMock and mixed mock modular forms in the lower half-plane
We study mock and mixed mock modular forms in the lower half-plane. In particular, our results apply to Zwegers’ three-variable mock Jacobi form μ(u, v; τ), three-variable generalizations of the universal mock modular partition rank generating function, and the quantum and mock modular strongly unimodal sequence rank generating function. We do not rely upon the analytic properties of these func...
متن کاملMock Jacobi Forms in Basic Hypergeometric Series
We show that some q-series such as universal mock theta functions are linear sums of theta quotient and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are multiplied by suitable powers of q. And we prove that certain linear sums of q-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or com...
متن کاملp-adic aspects of Jacobi forms
We are interested in understanding and describing the p-adic properties of Jacobi forms. As opposed to the case of modular forms, not much work has been done in this area. The literature includes [3, 4, 7]. In the first section, we follow Serre’s ideas from his theory of p-adic modular forms. We study Jacobi forms whose Fourier expansions have integral coefficients and look at congruences betwe...
متن کاملJACOBI’S TRIPLE PRODUCT, MOCK THETA FUNCTIONS, AND THE q-BRACKET
In Ramanujan’s final letter to Hardy, he wrote of a strange new class of infinite series he called “mock theta functions”. It turns out all of Ramanujan’s mock theta functions are essentially specializations of a so-called universal mock theta function g3(z, q) of Gordon–McIntosh. Here we show that g3 arises naturally from the reciprocal of the classical Jacobi triple product—and is intimately ...
متن کامل