Hecke-type double sums, Appell-Lerch sums, and mock theta functions, I

2014/03/20 by Dean Hickerson, Dean R. Hickerson, Eric T. Mortenson · 10 citations
Mathematics · #Advanced Mathematical Identities #Advanced Combinatorial Mathematics #Analytic Number Theory Research

paper · doi:10.1112/plms/pdu007

Abstract

By introducing a dual notion between partial theta functions and Appell–Lerch sums, we find and prove a formula which expresses Hecke-type double sums in terms of Appell–Lerch sums. Not only does our formula prove classical Hecke-type sum identities such as those in work of Kac and Peterson on affine Lie Algebras and Hecke modular forms, but once we have the Hecke-type forms for Ramanujan's mock theta functions our formula gives straightforward proofs of many of the classical mock theta identities. Indeed, we obtain a new proof of the mock theta conjectures. Our formula also applies to positive-level string functions associated with admissible representations of the affine Lie Algebra A 1 ( 1 ) of Kac and Wakimoto.

Citations

• The f(q) mock theta function conjecture and partition ranks
• Second Order Mock Theta Functions
• The Final Problem : An Account of the Mock Theta Functions
• Sixth order mock theta functions
• Ramanujan's “lost” notebook VII: The sixth order mock theta functions
• Tenth Order Mock Theta Functions in Ramanujan's Lost Notebook II
• Dyson’s ranks and Maass forms
• Some Eighth Order Mock Theta Functions
• Tenth order mock theta functions in Ramanujan's Lost Notebook

Cited by

• New polar-finite forms of generalized Euler identities for A₁⁽¹⁾-string functions and mock theta conjecture-like identities
• Renormalization and quantum modular forms, part II: Mock theta functions
• Four identities related to third order mock theta functions
• Eulerian series, zeta functions and the arithmetic of partitions
• Generalized rank deviations for overpartitions
• Some new mock theta functions
• Generalized Lambert Series Identities and Applications in Rank Differences
• Universal mock theta functions and two-variable Hecke-Rogers identities
• Bilateral Two-Parameter Mock Theta Functions and Related Applications
• Properties of Lerch Sums and Ramanujan's Mock Theta Functions
• Proofs of Two Conjectural Identities on Partial Nahm Sums
• Generating Functions of the Hurwitz Class Numbers Associated with Certain Mock Theta Functions

Related

• Representations of mock theta functions
• Two 2/5-level mock theta conjecture-like identities
• New polar-finite forms of generalized Euler identities for A₁⁽¹⁾-string functions and mock theta conjecture-like identities
• The H and K Family of Mock Theta Functions
• Properties of Lerch Sums and Ramanujan's Mock Theta Functions