Gaussian Hypergeometric Series and Combinatorial Congruences

2001 ◽  
pp. 1-12 ◽  
Author(s):  
Scott Ahlgren
Keyword(s):  
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.

scholarly journals Edwards Curves and Gaussian Hypergeometric Series

2016 ◽  
Vol 28 (1) ◽  
pp. 115-124 ◽  
Author(s):  
Mohammad Sadek ◽  
Nermine El-Sissi
Keyword(s):  
Hypergeometric Series ◽  
Edwards Curves ◽  
Gaussian Hypergeometric Series

scholarly journals The arithmetic of vector-valued modular forms on Γ0(2)

2019 ◽  
Vol 16 (02) ◽  
pp. 241-289
Author(s):  
Richard Gottesman
Keyword(s):  
Modular Forms ◽  
Hypergeometric Series ◽  
Fourier Coefficients ◽  
Weight Vector ◽  
Free Module ◽  
Two Dimensional ◽  
Dimensional Representation ◽  
Minimal Weight ◽  
Gaussian Hypergeometric Series ◽  
Vector Valued

Let [Formula: see text] denote an irreducible two-dimensional representation of [Formula: see text] The collection of vector-valued modular forms for [Formula: see text], which we denote by [Formula: see text], form a graded and free module of rank two over the ring of modular forms on [Formula: see text], which we denote by [Formula: see text] For a certain class of [Formula: see text], we prove that if [Formula: see text] is any vector-valued modular form for [Formula: see text] whose component functions have algebraic Fourier coefficients then the sequence of the denominators of the Fourier coefficients of both component functions of [Formula: see text] is unbounded. Our methods involve computing an explicit basis for [Formula: see text] as a [Formula: see text]-module. We give formulas for the component functions of a minimal weight vector-valued form for [Formula: see text] in terms of the Gaussian hypergeometric series [Formula: see text], a Hauptmodul of [Formula: see text], and the Dedekind [Formula: see text]-function.

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