A Gaussian hypergeometric series evaluation and Apéry number congruences

By Scott Ahlgren, Ken Ichi Ono. 2000

185 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e224c558dbb4417e0d3ef6e12c92152e6519cbfc

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

A p-adic analogue of a formula of Ramanujan (2007, background): “These series are analogous to classical hypergeometric series and have played an important role in relation to the number of points over Fp of Calabi-Yau threefolds [1], traces of Hecke operators [9], formulas for Ramanujan’s τ -function [19], and the number of points on a family of elliptic curves ”
Supercongruences for polynomial analogs of the Apéry numbers (2018, background): “Ono [3], who reduce it to the identity n ∑”
Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series (2018, methodology): “= [4N + 1][4N + 3](q; q(2))N+1(q; q (2))2N(q (3); q(6))N (1− q3)(q6; q(6))N(q(2); q(2))2N(q(6); q(2))N = [4N + 1][4N + 3][2N + 1][2][4] [3][2N + 2][2N + 4](−q; q)(2)N(−q; q)(2)2N(−q(3); q(3))(2)N [ 2N N ] [ 4N 2N ] [ 2N N ]”
Gaussian hypergeometric functions and traces of Hecke operators (2004, methodology): “12), the Gross-Koblitz formula for the p-adic Gamma-function, some p-adic analysis, and the WZ method, Ahlgren and the second author [3] successfully proved this conjecture.”
The trace of Frobenius of elliptic curves and the p-adic gamma function (2012, background): “Hypergeometric functions over finite fields have been applied to many areas but most interestingly perhaps has been their relationships to modular forms [1, 3, 5, 6, 14, 16, 17, 18] and their use in evaluating the number of points over Fp on certain algebraic varieties [1, 5, 14, 20].”
Hypergeometric Functions over Finite Fields (2015, methodology): “For the family C [5;3,4,4] λ , we have dimV1 = 2 and dimV2 = 1.”
Hypergeometric Series, Truncated Hypergeometric Series, and Gaussian Hypergeometric Functions (2015, methodology): “[15] Sharon Frechette, Ken Ono, and Matthew Papanikolas.”
Hypergeometric functions over F_p and relations to elliptic curves and modular forms (2008, background): “Acknowledgments The author thanks her advisor M. Papanikolas for his advice and support during the preparation of this paper.”
Congruences concerning Legendre polynomials (2010, background): “It is well known that (see [1], [10]) A( p−1 2 ) ≡ a(p) (mod p(2)), where a(n) is defined by q ∞ ∏”
On a supercongruence conjecture of Rodriguez-Villegas (2012, methodology): “The main approach for proving these types of supercongruences has been to use congruence relations between truncated generalized hypergeometric series and Greene’s hypergeometric functions over finite fields [1, 2, 11, 16, 18, 19, 20].”