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.

中文翻译：

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基于矩阵求逆的一般q展开式及其应用

本文采用矩阵求逆的方法，针对基数建立了任意形式幂级数F（z）的一般q展开式

$$ \ begin {aligned} \ left \ {z ^ n \ frac {（az）_ {n}} {（bz）_ {n}} \ bigg | n = 0,1,2，\ ldots \ right \ }。\ end {aligned} $$

给出了一些具体的展开式及其在q系列恒等式中的应用，包括Carlitz的q-展开式和新的偏theta函数恒等式以及Ramanujan的\（{{_ 1 \ psi _1 \））求和公式的系数恒等式案件。