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Transformations of basic hypergeometric series from the viewpoint of substitution of parameter operators

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Abstract

In this paper, we propose an operator method arising from the usual substitution of parameters to transformation formulas of basic hypergeometric series. As applications, some concrete substitutions of parameter operators implied by some known and fundamental transformation formulas are defined, thereby applications to the \({}_{r+1}\phi _{r}\) series for \(r=1, 2\) are exploited. Among these applications, three results are of interest: one is an observation that Heine’s transformations are both finite and closed under composition operation; the second one is the equivalency of Watson’s transformation of \({}_2\phi _1\) series and Weierstrass’ well-known theta function identity; and the third one is a generalization of Morita’s connection formula for two independent solutions to the q-confluent hypergeometric equation.

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Acknowledgements

The first author is indebted to Professor B. C. Berndt for offering him a visiting chance from July 15, 2015 to January 30, 2016, and to U of I at Urbana-Champaign for comprehensive library and computer conditions. During this period, the present paper evolved. We are indebted to the anonymous referee(s) for some extremely useful comments and criticisms on the results of this paper thereby helping us improve this manuscript. Thanks are also due to Professor S. O. Warnaar for drawing our attention to Tom H. Koornwinder’s paper on Weierstrass’ theta function identity.

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  1. Department of Mathematics, Soochow University, Suzhou, 215006, People’s Republic of China

    Xinrong Ma & Jin Wang

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  1. Xinrong Ma
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  2. Jin Wang
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Correspondence to Xinrong Ma.

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This work was supported by the National Natural Science Foundation of China [Grant No. 11471237].

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Ma, X., Wang, J. Transformations of basic hypergeometric series from the viewpoint of substitution of parameter operators. Ramanujan J 50, 61–91 (2019). https://doi.org/10.1007/s11139-019-00145-0

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