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Michael J. Schlosser - One of the best experts on this subject based on the ideXlab platform.

Wenchang Chu - One of the best experts on this subject based on the ideXlab platform.

Yasushi Kajihara - One of the best experts on this subject based on the ideXlab platform.

Rupam Barman - One of the best experts on this subject based on the ideXlab platform.

  • Certain product formulas and values of Gaussian Hypergeometric Series
    Research in Number Theory, 2020
    Co-Authors: Mohit Tripathi, Rupam Barman
    Abstract:

    In this article we find finite field analogues of certain product formulas satisfied by the classical Hypergeometric Series. We express product of two $${_2}F_1$$ 2 F 1 -Gaussian Hypergeometric Series as $${_4}F_3$$ 4 F 3 - and $${_3}F_2$$ 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 $${_4}F_3$$ 4 F 3 - and $${_3}F_2$$ 3 F 2 -Gaussian Hypergeometric Series. By counting points on CM elliptic curves over finite fields, Ono found certain special values of $${_2}F_1$$ 2 F 1 - and $${_3}F_2$$ 3 F 2 -Gaussian Hypergeometric Series containing trivial and quadratic characters as parameters. Later, Evans and Greene found special values of certain $${_3}F_2$$ 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.

  • Summation identities and transformations for Hypergeometric Series
    Annales mathématiques du Québec, 2018
    Co-Authors: Rupam Barman, Neelam Saikia
    Abstract:

    Nous trouvons des identités et des transformations de sommations pour les séries p -adiques hypergéométriques de McCarthy en évaluant certaines sommes de Gauss qui apparaissent lorsque nous comptons le nombre de points sur un corps fini $$\mathbb {F}_p$$ F p de la famille $$\begin{aligned} Z_{\lambda } : x_1^d+x_2^d = d\lambda x_1x_2^{d-1}. \end{aligned}$$ Z λ : x 1 d + x 2 d = d λ x 1 x 2 d - 1 . Pour sa part, Salerno exprime le nombre de points sur un corps fini $$ \mathbb {F}_p$$ F p de la famille $$Z_{\lambda }$$ Z λ en termes de quotients de fonctions gamma p -adiques sous la condition que d divise $$p-1$$ p - 1 . Dans cet article, nous exprimons d’abord le nombre de points sur un corps fini $$\mathbb {F}_p$$ F p de la famille $$Z_{\lambda }$$ Z λ en termes de séries hypergéométriques p -adiques de McCarthy pour tout nombre premier impair p ne divisant pas $$d(d-1)$$ d ( d - 1 ) , et déduisons ensuite deux identités de sommations pour les séries hypergéométriques p -adiques. Nous trouvons aussi certaines transformations et des valeurs spéciales de séries hypergéométriques. Finalement, nous trouvons une identité de sommations pour les séries hypergéométriques sur un corps fini de Greene. We find summation identities and transformations for the McCarthy’s p -adic Hypergeometric Series by evaluating certain Gauss sums which appear while counting points on the family $$\begin{aligned} Z_{\lambda }: x_1^d+x_2^d=d\lambda x_1x_2^{d-1} \end{aligned}$$ Z λ : x 1 d + x 2 d = d λ x 1 x 2 d - 1 over a finite field $$\mathbb {F}_p$$ F p . Salerno expresses the number of points over a finite field $$\mathbb {F}_p$$ F p on the family $$Z_{\lambda }$$ Z λ in terms of quotients of p -adic gamma functions under the condition that $$d|p-1$$ d | p - 1 . In this paper, we first express the number of points over a finite field $$\mathbb {F}_p$$ F p on the family $$Z_{\lambda }$$ Z λ in terms of McCarthy’s p -adic Hypergeometric Series for any odd prime p not dividing $$d(d-1)$$ d ( d - 1 ) , and then deduce two summation identities for the p -adic Hypergeometric Series. We also find certain transformations and special values of the p -adic Hypergeometric Series. We finally find a summation identity for the Greene’s finite field Hypergeometric Series.

  • Certain character sums and Hypergeometric Series
    arXiv: Number Theory, 2018
    Co-Authors: Rupam Barman, Neelam Saikia
    Abstract:

    We prove two transformations for the $p$-adic Hypergeometric Series which can be described as $p$-adic analogues of a Kummer's linear transformation and a transformation of Clausen. We first evaluate two character sums, and then relate them to the $p$-adic Hypergeometric Series to deduce the transformations. We also find another transformation for the $p$-adic Hypergeometric Series from which many special values of the $p$-adic Hypergeometric Series as well as finite field Hypergeometric functions are obtained.

  • Summation identities and transformations for Hypergeometric Series
    Annales mathématiques du Québec, 2017
    Co-Authors: Rupam Barman, Neelam Saikia
    Abstract:

    We find summation identities and transformations for the McCarthy’s p-adic Hypergeometric Series by evaluating certain Gauss sums which appear while counting points on the family $$\begin{aligned} Z_{\lambda }: x_1^d+x_2^d=d\lambda x_1x_2^{d-1} \end{aligned}$$ over a finite field $$\mathbb {F}_p$$ . Salerno expresses the number of points over a finite field $$\mathbb {F}_p$$ on the family $$Z_{\lambda }$$ in terms of quotients of p-adic gamma functions under the condition that $$d|p-1$$ . In this paper, we first express the number of points over a finite field $$\mathbb {F}_p$$ on the family $$Z_{\lambda }$$ in terms of McCarthy’s p-adic Hypergeometric Series for any odd prime p not dividing $$d(d-1)$$ , and then deduce two summation identities for the p-adic Hypergeometric Series. We also find certain transformations and special values of the p-adic Hypergeometric Series. We finally find a summation identity for the Greene’s finite field Hypergeometric Series.

  • Summation identities and transformations for Hypergeometric Series
    arXiv: Number Theory, 2016
    Co-Authors: Rupam Barman, Neelam Saikia
    Abstract:

    We find summation identities and transformations for the McCarthy's $p$-adic Hypergeometric Series by evaluating certain Gauss sums which appear while counting points on the family $$Z_{\lambda}: x_1^d+x_2^d=d\lambda x_1x_2^{d-1}$$ over a finite field $\mathbb{F}_p$. A. Salerno expresses the number of points over a finite field $\mathbb{F}_p$ on the family $Z_{\lambda}$ in terms of quotients of $p$-adic gamma function under the condition that $d|p-1$. In this paper, we first express the number of points over a finite field $\mathbb{F}_p$ on the family $Z_{\lambda}$ in terms of McCarthy's $p$-adic Hypergeometric Series for any odd prime $p$ not dividing $d(d-1)$, and then deduce two summation identities for the $p$-adic Hypergeometric Series. We also find certain transformations and special values of the $p$-adic Hypergeometric Series. We finally find a summation identity for the Greene's finite field Hypergeometric Series.

Hjalmar Rosengren - One of the best experts on this subject based on the ideXlab platform.