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Appell series over finite fields and Gaussian hypergeometric series

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Abstract

In this article, we find finite field analogues of certain identities satisfied by the classical \({_4}F_3\)-hypergeometric series and Appell series. As an application, we find new summation and product formulas satisfied by the Gaussian hypergeometric series. For example, we express a \({_4}F_3\)-Gaussian hypergeometric series as a sum of two \({_2}F_1\)-Gaussian hypergeometric series. We also find two identities expressing \({_4}F_3\)-Gaussian hypergeometric series as a product of two \({_2}F_1\)-Gaussian hypergeometric series. As another application, we find a special value of a \({_4}F_3\)-Gaussian hypergeometric series using the summation formula.

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Acknowledgements

We thank the anonymous referees for their comments and suggestions. The second author is partially supported by a research grant MTR/2017/000067 under the MATRICS scheme of SERB, Department of Science and Technology, Government of India.

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Correspondence to Mohit Tripathi.

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Tripathi, M., Barman, R. Appell series over finite fields and Gaussian hypergeometric series. Res Math Sci 8, 28 (2021). https://doi.org/10.1007/s40687-021-00266-3

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