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A generalization of Zwegers' multivariable $μ$-function

G. Shibukawa, S. Tsuchimi

TL;DR

This work generalizes Zwegers' multivariable $\mu$-function by applying a one-parameter deformation via iterative $q$-Borel summation, introducing the multivariable generalized $\mu$-function $\hat{\mu}_N$ and its fundamental relation to a higher-order $q$-difference equation. It constructs and analyzes a vector-valued companion $M_N$ and its modular completion, establishing explicit modular transformation formulas that involve Mordell integrals $h$ and correction terms $R_N$. The authors also derive detailed modular transformations for the multivariable $\mu$-function itself, treating odd and even $N$ separately, and demonstrate that the completed forms satisfy precise modular properties. The results deepen the connection between $q$-difference equations, $q$-Borel summation, and the modular theory of multivariate mock- or quantum-modular objects, with potential applications in the study of mock theta phenomena in higher dimensions.

Abstract

We introduce a one parameter deformation of Zwegers' multivariable $μ$-function by applying iterations of the $q$-Borel summation method, which is also a multivariate analogue of the generalized $μ$-function introduced by the authors. For this deformed multivariable $μ$-function, we give some formulas, for example, forward shift formula, translation and $\mathfrak{S}_{N+1}$-symmetry. Further we mention modular formulas for the Zwegers' original multivariable $μ$-function.

A generalization of Zwegers' multivariable $μ$-function

TL;DR

This work generalizes Zwegers' multivariable -function by applying a one-parameter deformation via iterative -Borel summation, introducing the multivariable generalized -function and its fundamental relation to a higher-order -difference equation. It constructs and analyzes a vector-valued companion and its modular completion, establishing explicit modular transformation formulas that involve Mordell integrals and correction terms . The authors also derive detailed modular transformations for the multivariable -function itself, treating odd and even separately, and demonstrate that the completed forms satisfy precise modular properties. The results deepen the connection between -difference equations, -Borel summation, and the modular theory of multivariate mock- or quantum-modular objects, with potential applications in the study of mock theta phenomena in higher dimensions.

Abstract

We introduce a one parameter deformation of Zwegers' multivariable -function by applying iterations of the -Borel summation method, which is also a multivariate analogue of the generalized -function introduced by the authors. For this deformed multivariable -function, we give some formulas, for example, forward shift formula, translation and -symmetry. Further we mention modular formulas for the Zwegers' original multivariable -function.

Paper Structure

This paper contains 7 sections, 21 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Proofs of Theorem \ref{['thm: mul mua']} and \ref{['prop: MN']}
  3. Proofs of Theorem \ref{['thm: MN modular']} and Theorem \ref{['thm: modular completion']}
  4. Modular transformations of the multivariable $\mu$-function
  5. The case of odd $N$
  6. The case of even $N$
  7. A connection problem of the $q$-difference equation \ref{['multi gmu eq']}

Key Result

Theorem 1

For any $r,s\in\{0,\ldots,N\}$ and $\sigma \in \mathfrak{S}_{N+1}$, then we have where $\mathfrak{S}_{N+1}$ is the symmetric group of the degree $N+1$.

Theorems & Definitions (41)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Theorem 4: Zw2
  • Lemma 1
  • Remark 1
  • Theorem 5
  • ...and 31 more