# A general q-expansion formula based on matrix inversions and its applications

## TL;DR

This paper introduces a general q-expansion formula using matrix inversions, enabling new identities and applications in q-series, including special cases like Carlitz's formula and Ramanujan's summation.

## Contribution

It develops a unified q-expansion formula based on matrix inversions, broadening the scope of q-series identities and applications.

## Key findings

- Derived a general q-expansion formula for formal power series.
- Presented new applications to q-series identities and special functions.
- Revealed connections to classical formulas like Carlitz's and Ramanujan's summation.

## Abstract

In this paper, by use of matrix inversions, we establish a general $q$-expansion formula of arbitrary formal power series $F(z)$ with respect to the base $$\left\{z^n\frac{(az:q)_n}{(bz:q)_n}\bigg|n=0,1,2\cdots\right\}.$$ Some concrete expansion formulas and their applications to $q$-series identities are presented, including Carlitz's $q$-expansion formula, a new partial theta function identity and a coefficient identity of Ramanujan's ${}_1\psi_1$ summation formula as special cases.