Abstract
In this article we find finite field analogues of certain product formulas satisfied by the classical hypergeometric series. We express product of two \({_2}F_1\)-Gaussian hypergeometric series as \({_4}F_3\)- and \({_3}F_2\)-Gaussian hypergeometric series. We use properties of Gauss and Jacobi sums and our earlier works on finite field Appell series to deduce these product formulas satisfied by the Gaussian hypergeometric series. We then use these transformations to evaluate explicitly some special values of \({_4}F_3\)- and \({_3}F_2\)-Gaussian hypergeometric series. By counting points on CM elliptic curves over finite fields, Ono found certain special values of \({_2}F_1\)- and \({_3}F_2\)-Gaussian hypergeometric series containing trivial and quadratic characters as parameters. Later, Evans and Greene found special values of certain \({_3}F_2\)-Gaussian hypergeometric series containing arbitrary characters as parameters from where some of the values obtained by Ono follow as special cases. We show that some of the results of Evans and Greene follow from our product formulas including a finite field analogue of the classical Clausen’s identity.
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References
Ahlgren, S., Ono, K.: A Gaussian hypergeometric series evaluation and Apéry number congruences. J. Reine Angew. Math. 518, 187–212 (2000)
Bailey, W.: Some theorem concerning products of hypergeometric series. Proc. Lond. Math. Soc. 28, 377–384 (1933)
Barman, R., Kalita, G.: Certain values of Gaussian hypergeometric series and a family of algebraic curves. Int. J. Number Theory 8(4), 945–961 (2012)
Barman, R., Kalita, G.: Elliptic curves and special values of Gaussian hypergeometric series. J. Number Theory 133(9), 3099–3111 (2013)
Berndt, B., Evans, R., Williams, K.: Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1998)
Evans, R.: Some mixed character sum identities of Katz. J. Number Theory 179, 17–32 (2017)
Evans, R., Greene, J.: Clausen’s theorem and hypergeometric functions over finite fields. Finite Fields Appl. 15(1), 97–109 (2009)
Evans, R., Greene, J.: Evaluations of hypergeometric functions over finite fields. Hiroshima Math. J. 39(2), 217–235 (2009)
Evans, R., Greene, J.: A quadratic hypergeometric \({_2}F_1\) transformation over finite field. Proc. Am. Math. Soc. 145, 1071–1076 (2017)
Evans, R., Greene, J.: Some mixed character sum identities of Katz II. Res. Number Theory 3, 8 (2017)
Fuselier, J., Long, L., Ramakrishna, R., Swisher, H., Tu, F.: Hypergeometric functions over finite fields. arXiv:1510.02575v3 (2017)
Greene, J.: Character sum analogues for hypergeometric and generalized hypergeometric functions over finite fields. Univ. of Minnesota, Minneapolis (1984). Ph. D. thesis
Greene, J.: Hypergeometric functions over finite fields. Trans. Am. Math. Soc. 301(1), 77–101 (1987)
Greene, J., Stanton, D.: A character sum evaluation and Gaussian hypergeometric series. J. Number Theory 23(1), 136–148 (1986)
Kalita, G.: Values of Gaussian hypergeometric series and their connections to algebraic curves. Int. J. Number Theory 14(1), 1–18 (2018)
Katz, N.: Exponential Sums and Differential Equations. Annals of Mathematics Studies. Princeton University Press, Princeton, NJ (1990)
Lang, S.: Cyclotomic Fields I and II, Graduate Texts in Mathematics, vol. 121. Springer, New York (1990)
McCarthy, D.: Transformations of well-poised hypergeometric functions over finite fields. Finite Fields Appl. 18(6), 1133–1147 (2012)
McCarthy, D., Papanicolas, M.A.: A finite field hypergeometric function associated to eigenvalues of a Siegel eigenform. Int. J. Number Theory 11(8), 2431–2450 (2015)
Ono, K.: Values of Gaussian hypergeometric series. Trans. Am. Math. Soc 350, 1205–1223 (1998)
Sadek, M., El-Sissi, N., Shamsi, A., Zamani, N.: Evaluation of Gaussian hypergeometric series using Huff’s models of elliptic curves. Ramanujan J. 48(2), 357–368 (2019)
Tripathi, M., Barman, R.: Certain transformations and special values of hypergeometric functions over finite fields. (under review)
Tripathi, M., Barman, R.: A finite field analogue of the Appell series \(F_4\). Res. Number Theory 4, 35 (2018)
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We thank the anonymous referee for his/her thorough review and highly appreciate the comments and suggestions, which significantly contributed to improving the quality of the paper. The second author is partially supported by a research grant under the MATRICS scheme of SERB, Department of Science and Technology , Government of India
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Tripathi, M., Barman, R. Certain product formulas and values of Gaussian hypergeometric series. Res. number theory 6, 26 (2020). https://doi.org/10.1007/s40993-020-00203-3
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DOI: https://doi.org/10.1007/s40993-020-00203-3