Appell series over finite fields and Gaussian hypergeometric series

2021 ◽  
Vol 8 (2) ◽  
Author(s):  
Mohit Tripathi ◽  
Rupam Barman
Keyword(s):  
Finite Fields ◽  
Hypergeometric Series ◽  
Gaussian Hypergeometric Series

Special values of the hypergeometric series II

2001 ◽  
Vol 131 (2) ◽  
pp. 309-319 ◽  
Author(s):  
I. J. ZUCKER ◽  
G. S. JOYCE
Keyword(s):  
Gamma Function ◽  
Hypergeometric Series ◽  
Elliptic Integral ◽  
Singular Values ◽  
Complete Elliptic Integral ◽  
Special Values ◽  
Gaussian Hypergeometric Series

Several authors [1, 5, 9] have investigated the algebraic and transcendental values of the Gaussian hypergeometric series(formula here)for rational parameters a, b, c and algebraic and rational values of z ∈ (0, 1). This led to several new identities such as(formula here)and(formula here)where Γ(x) denotes the gamma function. It was pointed out by the present authors [6] that these results, and others like it, could be derived simply by combining certain classical F transformation formulae with the singular values of the complete elliptic integral of the first kind K(k), where k denotes the modulus.Here, we pursue the methods used in [6] to produce further examples of the type (1·2) and (1·3). Thus, we find the following results:(formula here)The result (1·6) is of particular interest because the argument and value of the F function are both rational.

scholarly journals Gaussian Hypergeometric series and supercongruences

2009 ◽  
Vol 78 (265) ◽  
pp. 275-275 ◽  
Author(s):  
Robert Osburn ◽  
Carsten Schneider
Keyword(s):  
Hypergeometric Series ◽  
Gaussian Hypergeometric Series

Hyperelliptic curves and values of Gaussian hypergeometric series

2014 ◽  
Vol 102 (4) ◽  
pp. 345-355 ◽  
Author(s):  
Rupam Barman ◽  
Gautam Kalita ◽  
Neelam Saikia
Keyword(s):  
Hypergeometric Series ◽  
Hyperelliptic Curves ◽  
Gaussian Hypergeometric Series

scholarly journals 3F2 HYPERGEOMETRIC SERIES AND PERIODS OF ELLIPTIC CURVES

2010 ◽  
Vol 06 (03) ◽  
pp. 461-470 ◽  
Author(s):  
DERMOT McCARTHY
Keyword(s):  
Elliptic Curves ◽  
Hypergeometric Series ◽  
The Real ◽  
Gaussian Hypergeometric Series ◽  
Trace Of Frobenius

We express the real period of a family of elliptic curves in terms of classical hypergeometric series. This expression is analogous to a result of Ono which relates the trace of Frobenius of the same family of elliptic curves to a Gaussian hypergeometric series. This analogy provides further evidence of the interplay between classical and Gaussian hypergeometric series.

Values of Gaussian hypergeometric series and their connections to algebraic curves

2017 ◽  
Vol 14 (01) ◽  
pp. 1-18 ◽  
Author(s):  
Gautam Kalita
Keyword(s):  
Hypergeometric Series ◽  
Algebraic Curves ◽  
Vital Role ◽  
Special Values ◽  
Transformation Formulas ◽  
Gaussian Hypergeometric Series

In this paper, we explicitly evaluate certain special values of [Formula: see text] hypergeometric series. These evaluations are based on some summation transformation formulas of Gaussian hypergeometric series. We find expressions of the number of points on certain algebraic curves over [Formula: see text] in terms of Gaussian hypergeometric series, which play the vital role in deducing the transformation results.

Appell series 𝔽1 over finite fields

2018 ◽  
Vol 14 (03) ◽  
pp. 727-738 ◽  
Author(s):  
Long Li ◽  
Xin Li ◽  
Rui Mao
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Generating Functions ◽  
Hypergeometric Series

In 1987, Greene introduced the notion of the finite field analogue of hypergeometric series. In this paper, we give a finite field analogue of Appell series and obtain some transformation and reduction formulas. We also establish the generating functions for Appell series over finite fields.

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