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\({}_{4}F_{3}\)-Gaussian hypergeometric series and traces of Frobenius for elliptic curves

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Abstract

In this article, we obtain finite field analogues of classical summation identities connecting \(F_3\)-Appell series and \({_4} F_3\)-classical hypergeometric series. As an application, we establish a new summation formula satisfied by the \({_4} F_3\)-Gaussian hypergeometric series. We further express certain special values of \({_4} F_3\)-Gaussian hypergeometric series in terms of traces of the Frobenius endomorphisms of certain families of elliptic curves. We also explicitly find some special values of \({{}_4} F_3\)-Gaussian hypergeometric series.

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Acknowledgements

The research was partially supported by the DST-SERB grant CRG/2020/004147. We would like to thank Prof. Ling Long for her useful suggestions.

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  1. School of Mathematical Sciences, National Institute of Science Education and Research, Bhubaneswar, An OCC of Homi Bhabha National Institute, P. O. Jatni, Khurda, Odisha, 752050, India

    Mohit Tripathi & Jaban Meher

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  1. Mohit Tripathi
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  2. Jaban Meher
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Correspondence to Mohit Tripathi.

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Tripathi, M., Meher, J. \({}_{4}F_{3}\)-Gaussian hypergeometric series and traces of Frobenius for elliptic curves. Res Math Sci 9, 63 (2022). https://doi.org/10.1007/s40687-022-00358-8

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