On the $q$-derivative and $q$-series expansions

TL;DR

This paper develops new $q$-series formulas involving infinite products, derives Hecke-type series identities, and provides novel representations for sums of squares and triangular numbers, connecting to classical results.

Contribution

It introduces a general $q$-series expansion method that yields new identities and representations related to sums of squares and triangular numbers.

Findings

Derived new $q$-formulas with infinite products

Established multiple Hecke-type series identities

Provided a new representation for sums of three triangular numbers

Abstract

Using a general $q$-series expansion, we derive some nontrivial $q$-formulas involving many infinite products. A multitude of Hecke--type series identities are derived. Some general formulas for sums of any number of squares are given. A new representation for the generating function for sums of three triangular numbers is derived, which is slightly different from that of Andrews, also implies the famous result of Gauss where every integer is the sum of three triangular numbers.