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

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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.

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Acknowledgements

The author is very indebted to the anonymous referee’s suggestion to investigate any combinatorial or number-theoretic interpretation of \(\{C_{n,k}^{(a,b)}\}_{n\ge k\ge 0}\), leading us to their relation with the Catalan numbers.

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  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Jin Wang

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  1. Jin Wang
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Correspondence to Jin Wang.

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This work was supported by the Natural Science Foundation of Zhejiang Province (Grant No. LQ20A010004) and by the National Natural Science Foundation of China (Grant Nos. 11471237 and 11971341).

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Wang, J. A general q-expansion formula based on matrix inversions and its applications. Ramanujan J 53, 493–516 (2020). https://doi.org/10.1007/s11139-020-00318-2

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  • DOI: https://doi.org/10.1007/s11139-020-00318-2

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