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A degeneration of the generalized Zwegers' $μ$-function according to the Ramanujan difference equation

G. Shibukawa, S. Tsuchimi

Abstract

In this paper, we introduce the little $μ$-function, which is obtained as a degenerate limit of the generalized $μ$-function. We derive the little $μ$-function as the image of the $q$-Borel summation of a divergent solution to the Ramanujan equation which is the most degenerate second order linear $q$-difference equations of Laplace type excluding those of constant coefficients. Moreover, we present several formulas, such as symmetries and connection formulas for the little $μ$-function, similar to those for the generalized $μ$-function. Furthermore, we establish contiguous relations related to the $q,t$-Fibonacci sequences and Wronskian relations involving the Rogers-Ramanujan continued fraction.

A degeneration of the generalized Zwegers' $μ$-function according to the Ramanujan difference equation

Abstract

In this paper, we introduce the little -function, which is obtained as a degenerate limit of the generalized -function. We derive the little -function as the image of the -Borel summation of a divergent solution to the Ramanujan equation which is the most degenerate second order linear -difference equations of Laplace type excluding those of constant coefficients. Moreover, we present several formulas, such as symmetries and connection formulas for the little -function, similar to those for the generalized -function. Furthermore, we establish contiguous relations related to the -Fibonacci sequences and Wronskian relations involving the Rogers-Ramanujan continued fraction.
Paper Structure (4 sections, 10 theorems, 83 equations, 6 figures)

This paper contains 4 sections, 10 theorems, 83 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Proofs of the main results
  3. Variations on the Ramanujan Equation
  4. Explicit formulas of $q,t$-Fibonacci sequences

Key Result

Theorem 1

We have the following equations:

Figures (6)

  • Figure 1: $v_1$
  • Figure 2: $v_2$
  • Figure 3: $v_3$
  • Figure 4: $v_4$
  • Figure 5: $v_5$
  • ...and 1 more figures

Theorems & Definitions (21)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 1
  • proof : Proof of Theorem $\ref{['thm: lmu and gmu']}$
  • Remark 1
  • proof : Proof of Theorem $\ref{['thm: lmu property']}$
  • Remark 2
  • ...and 11 more