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

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 $$\{z^n\frac{(az:q{)}_n}{(bz:q{)}_n}\mid n=0,1,2\cdots \}\mathrm{.}$$ 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.