Certain product formulas and values of Gaussian hypergeometric series
Mohit Tripathi,Rupam Barman +1 more
- Vol. 6, Iss: 3, pp 1-29
TLDR
In this paper, a finite field analogue of certain product formulas satisfied by the classical hypergeometric series was found, where the authors used properties of Gauss and Jacobi sums and their earlier works on finite field Appell series to deduce these product formulas satisfying by the Gaussian hypergeometrical series.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.read more
Citations
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Journal ArticleDOI
Appell series over finite fields and Gaussian hypergeometric series
Mohit Tripathi,Rupam Barman +1 more
TL;DR: In this paper, the authors find finite field analogues of certain identities satisfied by the classical Gaussian hypergeometric series and Appell series and derive new summation and product formulas satisfying these identities.
Journal ArticleDOI
Product formulas for hypergeometric functions over finite fields
Noriyuki Otsubo,Takato Senoue +1 more
TL;DR: In this article , the authors established analogous formulas for generalized hypergeometric functions over finite fields, where the product formulas are known classically for generalized HOG functions over the complex numbers.
Appell-Lauricella hypergeometric functions over finite fields and algebraic varieties
TL;DR: In this article , the authors consider certain surfaces having a group action and compute the numbers of rational points associated with characters of the group, which will be expressed in terms of Appell-Lauricella functions over finite variables.
Journal ArticleDOI
$${}_{4}F_{3}$$-Gaussian hypergeometric series and traces of Frobenius for elliptic curves
Mohit Tripathi,Jaban Meher +1 more
Posted Content
Distribution of values of Gaussian hypergeometric functions
Ken Ono,Hasan Saad,Neelam Saikia +2 more
TL;DR: For the $_2F_1$ functions, the limiting distribution is semicircular, whereas the distribution for the $_3F_2$ functions is the more exotic {\it Batman} distribution as mentioned in this paper.
References
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Book
Cyclotomic Fields I and II
TL;DR: In this article, the Iwasawa theory of local units is used to explain the Stickelberger Ideals and Bernoulli Distributions of Gauss Sums as well as the ArtinSchreier Curve.
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Exponential sums and differential equations
TL;DR: In this article, the theory of linear differential equations in one complex variable with polynomial coefficients and one parameter families of exponential sums over finite fields is discussed. And the results about differential equations and their differential galois groups and one-parameter families of exponentially sums and their geometric monodromy groups are compared.
Journal ArticleDOI
Hypergeometric functions over finite fields
TL;DR: In this paper, the analogy between the character sum expansion of a complex-valued function over GF(q) and the power series expansion of an analytic function is exploited in order to develop an analogue for hypergeometric series over finite fields.
Journal ArticleDOI
A Gaussian hypergeometric series evaluation and Apéry number congruences
Scott Ahlgren,Ken Ono +1 more
TL;DR: In this article, the authors obtained a simple formula for 4F3(1)p, where p is the trivial character modulo p and φp is the Legendre symbol modulo φ p. For n > 2 the non-trivial values of n+1Fn(x)p have been difficult to obtain.
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