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

Abstract

In this paper, we introduce the little $\mu$-function, which is obtained as a degenerate limit of the generalized $\mu$-function. We derive the little $\mu$-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 $\mu$-function, similar to those for the generalized $\mu$-function. Furthermore, we establish contiguous relations related to the $q,t$-Fibonacci sequences and Wronskian relations involving the Rogers-Ramanujan continued fraction.