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Transformation of Third Order Mock Theta Functions and New $q$-Series Identities

Frank Garvan, Avi Mukhopadhyay

TL;DR

This work provides a $\text{Zwegers}$-style derivation of the modular transformations for the third-order mock theta functions $f(q)$ and $\omega(q)$ by leveraging Appell-Lerch sums and their completion $\widetilde{\mu}$, avoiding Watson's original transformation formulas. The authors express $h_2(\tau)$ in terms of $\widetilde{\mu}$ and derive a new identity for $\omega(q)$, implemented via explicit $3$-dissections of eta-quotients and $\mu$-sums. The resulting identities, labeled $NEWOMEGA$, $NEWOMEGA2$, and $NEWF$, connect mock theta functions to eta-quotients and Eichler integrals, and they generalize the known transformations while providing a new independent proof of Zwegers' results. The approach clarifies the structure behind the mock modularity of these functions and suggests further parallels between Ramanujan's original identities and modern modular-analytic frameworks.

Abstract

Ramanujan introduced mock theta functions in his last letter to G.H.Hardy. He provided examples and various relations between them. G.N.Watson found transformations for the third order mock theta functions $f(q)$ and $ω$(q). Zwegers in 2000 built on Watson's techniques to complete these mock theta functions and connected them to real analytic modular forms. We show how to derive these transformations using Lerch sums. To show the equivalence of the results involves some new $q$-series identities thus resulting in a new proof of Zwegers' theorem.

Transformation of Third Order Mock Theta Functions and New $q$-Series Identities

TL;DR

This work provides a -style derivation of the modular transformations for the third-order mock theta functions and by leveraging Appell-Lerch sums and their completion , avoiding Watson's original transformation formulas. The authors express in terms of and derive a new identity for , implemented via explicit -dissections of eta-quotients and -sums. The resulting identities, labeled , , and , connect mock theta functions to eta-quotients and Eichler integrals, and they generalize the known transformations while providing a new independent proof of Zwegers' results. The approach clarifies the structure behind the mock modularity of these functions and suggests further parallels between Ramanujan's original identities and modern modular-analytic frameworks.

Abstract

Ramanujan introduced mock theta functions in his last letter to G.H.Hardy. He provided examples and various relations between them. G.N.Watson found transformations for the third order mock theta functions and (q). Zwegers in 2000 built on Watson's techniques to complete these mock theta functions and connected them to real analytic modular forms. We show how to derive these transformations using Lerch sums. To show the equivalence of the results involves some new -series identities thus resulting in a new proof of Zwegers' theorem.
Paper Structure (14 sections, 11 theorems, 104 equations)

This paper contains 14 sections, 11 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. Zwegers' Completion of Appell-Lerch Series
  4. Some $q$-series identities
  5. Proof of main results
  6. $h_{2}(\tau)$ in terms of the $\widetilde{\mu}$ function
  7. Transforming $h_2$ and a new identity for $\omega(q)$
  8. Dissection of an eta-quotient
  9. Dissection of a sum
  10. Bringing everything together
  11. The Other half
  12. Another identity involving the $\omega$ function
  13. An identity for $f(q)$
  14. Conclusion

Key Result

Lemma 1.1

For $q=e^{2 \pi i \tau}$, $\tau \in \mathbb{H}$ we have with $\zeta_{n}=e^{2\pi i/n}, R(\tau)= 4\sqrt{3}\sqrt{-i\tau}(j_1(\tau)),-j_1(\tau)),j_3(\tau))^{T}$, where

Theorems & Definitions (16)

  • Lemma 1.1: Watson Wa1936
  • Lemma 1.2
  • Theorem 1.3: Zwegers Zw2002
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Theorem 3.1
  • ...and 6 more