Evaluation of Hecke–Rogers series and expansions of the rank function

A formula is established for the evaluation of double series of Hecke–Rogers type in terms of theta functions and Appell–Lerch functions. This formula is similar to others obtained previously by Hickerson and Mortenson, and by Mortenson and Zwegers. It is applied to the rank function, leading to an...

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Bibliographic Details
Published in:The Ramanujan journal Vol. 65; no. 4; pp. 1949 - 1970
Main Author: Bradley-Thrush, J. G.
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2024
Subjects:
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Number Theory
Hecke–Rogers series
Basic hypergeometric functions
Appell–Lerch series
33D15
Theta functions
ISSN:1382-4090, 1572-9303
Online Access:Get full text
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Summary:A formula is established for the evaluation of double series of Hecke–Rogers type in terms of theta functions and Appell–Lerch functions. This formula is similar to others obtained previously by Hickerson and Mortenson, and by Mortenson and Zwegers. It is applied to the rank function, leading to an expansion closely analogous to two of Garvan’s double series identities. Several identities involving third-order mock theta functions are obtained as special cases.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-024-00842-5