منابع مشابه
Values of Gaussian Hypergeometric Series
Let p be prime and let GF (p) be the finite field with p elements. In this note we investigate the arithmetic properties of the Gaussian hypergeometric functions 2F1(x) =2 F1 „ φ, φ | x « and 3F2(x) =3 F2 „ φ, φ, φ , | x « where φ and respectively are the quadratic and trivial characters of GF (p). For all but finitely many rational numbers x = λ, there exist two elliptic curves 2E1(λ) and 3E2(...
متن کاملGaussian Hypergeometric series and supercongruences
Let p be an odd prime. In 1984, Greene introduced the notion of hypergeometric functions over finite fields. Special values of these functions have been of interest as they are related to the number of Fp points on algebraic varieties and to Fourier coefficients of modular forms. In this paper, we explicitly determine these functions modulo higher powers of p and discuss an application to super...
متن کاملGaussian Hypergeometric Series and Extensions of Supercongruences
Let p be an odd prime. The purpose of this paper is to refine methods of Ahlgren and Ono [2] and Kilbourn [13] in order to prove a general mod p congruence for the Gaussian hypergeometric series n+1Fn(λ) where n is an odd positive integer. As a result, we extend three recent supercongruences. The first is a result of Ono and Ahlgren [2] on a supercongruence for Apéry numbers which was conjectur...
متن کاملGaussian Hypergeometric Series and Combinatorial Congruences
In a recent paper [A-O], the author and K. Ono study the “Gaussian” hypergeometric series 4F3(1)p over the finite field Fp. They describe relationships between values of these series, Fourier coefficients of modular forms, and the arithmetic of a certain algebraic variety. These relationships, together with tools from p-adic analysis and some unexpected combinatorial identities, lead to the pro...
متن کاملPolynomial series expansions for confluent and Gaussian hypergeometric functions
Based on the Hadamard product of power series, polynomial series expansions for confluent hypergeometric functions M(a, c; ·) and for Gaussian hypergeometric functions F (a, b; c; ·) are introduced and studied. It turns out that the partial sums provide an interesting alternative for the numerical evaluation of the functions M(a, c; ·) and F (a, b; c; ·), in particular, if the parameters are al...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 1998
ISSN: 0002-9947,1088-6850
DOI: 10.1090/s0002-9947-98-01887-x