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

Abstract

By developing a connection 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 and theta functions. Not only does our formula prove classical Hecke-type double sum identities such as those found in work 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 function identities. In particular, we obtain a new proof of the mock theta conjectures. Our formula also applies to positive-level string functions associated with admissable representations of the affine Lie Algebra $A_1^{(1)}$ as introduced by Kac and Wakimoto.

References in corpus (2)

• Mock Theta Functions
• Mixed mock modular q-series

Cited by in corpus (42)

• On the dual nature of partial theta functions and Appell-Lerch sums
• Dyson's Ranks and Appell-Lerch Sums
• On ranks and cranks of partitions modulo $4$ and $8$
• Ramanujan's radial limits and mixed mock modular bilateral $q$-hypergeometric series
• Exotic Bailey-Slater SPT-Functions II: Hecke-Rogers-Type Double Sums and Bailey Pairs From Groups A, C, E
• Mock theta double sums
• Parity of coefficients of mock theta functions
• A double-sum Kronecker-type identity
• On the tenth-order mock theta functions
• On string functions and double-sum formulas
• On Ramanujan's lost notebook and new tenth-order like identities for second-, sixth-, and eighth-order mock theta functions
• Renormalization and quantum modular forms, part II: Mock theta functions
• Eulerian series, zeta functions and the arithmetic of partitions
• Proofs of Some Conjectures of Chan on Appell-Lerch Sums
• The mixed mock modularity of certain duals of generalized quantum modular forms of Hikami and Lovejoy
• Eulerian series as modular forms revisited
• Deviation of the rank and crank modulo 11
• On Hecke-type double-sums and general string functions for the affine Lie algebra $A_1^{(1)}$
• Mock Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series
• A new Andrews--Crandall-type identity and the number of integer solutions to $x^2+2y^2+2z^2=n$
• Expressing $q$-series in terms of building blocks of Hecke-type double-sums
• Splitting Appell functions in terms of single quotients of theta functions
• Hecke-Rogers double-sums and false theta functions
• Orientation Reversal and the Chern-Simons Natural Boundary
• Line defect half-indices of $SU(N)$ Chern-Simons theories
• More on some Mock theta Double sums
• Universal mock theta functions and two-variable Hecke-Rogers identities
• Arithmetic Properties of Odd Ranks and $k$-Marked Odd Durfee Symbols
• Threefield identities and simultaneous representations of primes by binary quadratic forms
• On recursions for coefficients of mock theta functions
• A general formula for Hecke-type false theta functions
• Two $2/5$-level mock theta conjecture-like identities
• A $q$-multisum identity arising from finite chain ring probabilities
• Four identities related to third order mock theta functions
• Properties of Lerch Sums and Ramanujan's Mock Theta Functions
• New polar-finite forms of generalized Euler identities for $A_1^{(1)}$-string functions and mock theta conjecture-like identities
• Rank deviations for overpartitions
• Short proofs of Ramanujan-like identities for the eighth order mock theta function $V_0(q)$
• Generalized Lambert Series Identities and Applications in Rank Differences
• Unimodal sequences and mixed false theta functions
• Asymptotic distribution of odd-balanced unimodal sequences with rank congruent to a mod c
• Generating Functions of the Hurwitz Class Numbers Associated with Certain Mock Theta Functions