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

Abstract

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