Quantum Jacobi forms and sums of tails identities

摘要

Our results are on the interconnected topics of quantum Jacobi sums of tails identities, quantum Jacobi and mock Jacobi properties of two-variable q -hypergeometric series and partial theta functions, and two-variable q -hypergeometric generating functions for certain L -values by way of related asymptotic expansions. More specifically, we establish five two-variable quantum Jacobi sums of tails identities. As corollaries, we recover known one-variable quantum sum of tails identities due to Zagier, Andrews-Jiménez-Urroz-Ono, and more. Further, justifying the “quantum Jacobi” description of our two-variable sums of tails identities, we establish the quantum Jacobi and mock Jacobi properties of a number of two-variable q -hypergeometric series and partial Jacobi theta functions which appear in our two-variable sums of tails identities, inspired by related results of Zagier and Rolen-Schneider in the one-variable quantum modular setting. Finally, by establishing related asymptotic expansions, we realize generating functions for certain L -values in terms of two-variable q -hypergeometric series and Jacobi partial theta functions, inspired by earlier work in this direction by Andrews–Jiménez-Urroz–Ono for the Riemann ζ documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$zeta $$end{document} -function and Dirichlet and Hecke L -functions.