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The generalized Zwegers' $μ$-function and transformation formulas for the bilateral basic hypergeometric series

Genki Shibukawa, Satoshi Tsuchimi

Abstract

By applying Slater's transformation formulas for the bilateral basic hypergeometric series ${}_2ψ_{2}$, we derive three type translation formulas for the generalized Zwegers' $μ$-function (``continuous $q$-Hermite function'') which was introduced by Shibukawa--Tsuchimi (SIGMA, 2023). From some Bailey's transformation formula of ${}_2ψ_{2}$, we also give a formula for the expression of the generalized Zwegers' $μ$-function by some very-well-poised bilateral basic hypergeometric series ${}_4ψ_{8}$. As an application of this new expression formula for the generalized Zwegers' $μ$-function, we obtain some new Fourier expansions for the Weierstrass and Jacobi elliptic functions.

The generalized Zwegers' $μ$-function and transformation formulas for the bilateral basic hypergeometric series

Abstract

By applying Slater's transformation formulas for the bilateral basic hypergeometric series , we derive three type translation formulas for the generalized Zwegers' -function (``continuous -Hermite function'') which was introduced by Shibukawa--Tsuchimi (SIGMA, 2023). From some Bailey's transformation formula of , we also give a formula for the expression of the generalized Zwegers' -function by some very-well-poised bilateral basic hypergeometric series . As an application of this new expression formula for the generalized Zwegers' -function, we obtain some new Fourier expansions for the Weierstrass and Jacobi elliptic functions.
Paper Structure (11 sections, 7 theorems, 66 equations)

This paper contains 11 sections, 7 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Some transformation formulas for the series ${}_2\psi_{2}$ and bilateral basic hypergeometric expressions of $\mu(x,y;a)$
  3. Proofs of main results
  4. Degenerate Very-Well-Poised bilateral series expressions for Ramanujan's mock theta functions
  5. Order 2 theta functions
  6. Order 3 mock theta functions
  7. Order 5 mock theta functions
  8. Order 6 mock theta functions
  9. Order 7 mock theta functions
  10. Order 8 mock theta functions
  11. Order 10 mock theta functions

Key Result

Theorem 1

For the free parameters $x^{\prime}$ and $y^{\prime}$, we have

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Lemma 3: GR, S
  • Remark 4
  • Lemma 5
  • Lemma 6: AB, B1, GR
  • Remark 7
  • Lemma 8: [ST, Theorem 1.4]
  • Remark 9
  • Corollary 10
  • ...and 2 more