Eulerian series as Hecke-type double sums through the transformation formulas

Abstract

Recently, establishing the Hecke-type double sums forms for Eulerian series has received a great deal of attention to seek special cases of [Formula: see text]-hypergeometric series. In this paper, with the aid of some basic hypergeometric series identities given by Chen and Wang, and Liu, we discover many new Eulerian series and express them in terms of Hecke-type double sums and Appell–Lerch sums. Based on the modern sense for the mock theta functions, we find that some of these series are mock theta functions and the others are mixed mock theta functions. Furthermore, from the viewpoint of Hecke-type double sums, we find three string function identities with levels [Formula: see text], [Formula: see text], and [Formula: see text].