Bilateral Two-Parameter Mock Theta Functions and Related Applications

TL;DR

The paper develops a framework for bilateral two-parameter mock theta functions by extending classical mock theta functions to bilateral series and expressing them through Appell–Lerch sums. It derives explicit identities linking bilateral versions of families such as $g_{3,c}(x,q)$ and $R_c(x,q)$ to their classical mock theta counterparts, including specialized relations that connect orders across the Ramanujan spectrum (e.g., third-, fifth-, and sixth-order functions). Key results are presented as main theorems (bg3-1, 3+2, 5+1, 0+5, 1+6) with proofs based on Appell–Lerch theory, and supplemented by alternative proofs in Appendices using bilateral hypergeometric techniques and Andrews’ lemma. The work expands understanding of how bilateral series encode modular-like structure, provides a catalog of bilateral identities, and suggests combinatorial interpretations through two-parameter generating functions such as $R(x,q)$ and $g_3(x,q)$.

Abstract

In this paper, we investigate new relationships for bilateral series related to two-parameter mock theta functions, which lead to many identities concerning the bilateral mock theta functions. In addition, interesting relations between the classical mock theta functions and the bilateral series are also concluded.

Theorems & Definitions (16)

• Theorem 2.1
• Corollary 2.2
• Theorem 2.3
• Corollary 2.4
• Corollary 2.5
• Theorem 2.6
• Corollary 2.7
• Corollary 2.8
• Theorem 2.9
• Corollary 2.10