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.

scholarly journals CERTAIN VALUES OF GAUSSIAN HYPERGEOMETRIC SERIES AND A FAMILY OF ALGEBRAIC CURVES

2012 ◽  
Vol 08 (04) ◽  
pp. 945-961 ◽  
Author(s):  
RUPAM BARMAN ◽  
GAUTAM KALITA
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Algebraic Curve ◽  
Hypergeometric Series ◽  
Algebraic Curves ◽  
Alternate Proof ◽  
Special Values ◽  
Gaussian Hypergeometric Series

Let λ ∈ ℚ\{0, -1} and l ≥ 2. Denote by Cl, λ the nonsingular projective algebraic curve over ℚ with affine equation given by [Formula: see text] In this paper, we give a relation between the number of points on Cl, λ over a finite field and Gaussian hypergeometric series. We also give an alternate proof of a result of [D. McCarthy, 3F2 Hypergeometric series and periods of elliptic curves, Int. J. Number Theory6(3) (2010) 461–470]. We find some special values of 3F2 and 2F1 Gaussian hypergeometric series. Finally we evaluate the value of 3F2(4) which extends a result of [K. Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc.350(3) (1998) 1205–1223].

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.

ON THE POLYNOMIAL xd + ax + b OVER 𝔽q AND GAUSSIAN HYPERGEOMETRIC SERIES

2013 ◽  
Vol 09 (07) ◽  
pp. 1753-1763 ◽  
Author(s):  
RUPAM BARMAN ◽  
GAUTAM KALITA
Keyword(s):  
Modular Forms ◽  
Hypergeometric Series ◽  
American Mathematical Society ◽  
Polynomial Equation ◽  
Mathematical Society ◽  
Number Of Solutions ◽  
Special Values ◽  
Regional Conference ◽  
Gaussian Hypergeometric Series ◽  
Cbms Regional Conference Series

For d ≥ 2, denote by Pd(x) the polynomial over 𝔽q given by [Formula: see text]. We explicitly find the number of solutions in 𝔽q of the polynomial equation Pd(x) = 0 in terms of special values of dFd-1 and d-1Fd-2 Gaussian hypergeometric series with characters of orders d and d - 1 as parameters. This solves a problem posed by K. Ono (see p. 204 in [Web of Modularity : Arithmetic of the Coefficients of Modular Forms and q-Series, CBMS Regional Conference Series in Mathematics, No. 102 (American Mathematical Society, Providence, RI, 2004)]) on special values of n+1Fn Gaussian hypergeometric series for n > 2.

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

Corrigendum

1991 ◽  
Vol 110 (3) ◽  
pp. 599-599
Keyword(s):  
Hypergeometric Series ◽  
Physics Laboratory ◽  
Special Values ◽  
King's College

Volume 109 (1991), 257–261’Special values of the hypergeometric series’By B G. S. Joyce and I. J. ZuckerWheatstone Physics Laboratory, King's College, Strand, London WC2R 2LSThe editor regrets that the errors listed below appeared in this paperp. 259, Table 1. In the last line π−1/21 should read π−1/2.p. 260. The k in equation (22) should be k.p. 261. The expression (am2 + bmn + cn2)−s should be (am2 + bmn + cn2)−s in reference[11].

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