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12 pages, 268 KiB

Open AccessArticle

Some New q-Congruences for Truncated Basic Hypergeometric Series

by Victor J. W. Guo and Michael J. Schlosser

Symmetry 2019, 11(2), 268; https://doi.org/10.3390/sym11020268 - 20 Feb 2019

Cited by 27 | Viewed by 3453

We provide several new q-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric generalizations thereof. These are established by a variety of techniques including polynomial argument, creative microscoping (a method recently introduced by the first author in collaboration with Zudilin), Andrews’ multiseries generalization of the Watson transformation, and induction. We also give a number of related conjectures including congruences modulo the fourth power of a cyclotomic polynomial. Full article

22 pages, 301 KiB

Open AccessArticle

ρ — Adic Analogues of Ramanujan Type Formulas for 1/π

by Sarah Chisholm, Alyson Deines, Ling Long, Gabriele Nebe and Holly Swisher

Mathematics 2013, 1(1), 9-30; https://doi.org/10.3390/math1010009 - 13 Mar 2013

Cited by 8 | Viewed by 6471

Abstract

Following Ramanujan's work on modular equations and approximations of

π

, there are formulas for

1 / π

of the form Following Ramanujan's work on modular equations and approximations of

π

, there are formulas for

1 / π

of the form [...] Read more.

Following Ramanujan's work on modular equations and approximations of

π

, there are formulas for

1 / π

of the form Following Ramanujan's work on modular equations and approximations of

π

, there are formulas for

1 / π

of the form

\sum_{k = 0}^{\infty} \frac{{(\frac{1}{2})}_{k} {(\frac{1}{d})}_{k} {(\frac{d - 1}{d})}_{k}}{k!^{3}} (a k + 1) {(λ_{d})}^{k} = \frac{δ}{π}

for

d = 2, 3, 4, 6,

where

ł_{d}

are singular values that correspond to elliptic curves with complex multiplication, and

a, δ

are explicit algebraic numbers. In this paper we prove a

p -

adic version of this formula in terms of the so-called Ramanujan type congruence. In addition, we obtain a new supercongruence result for elliptic curves with complex multiplication. Full article

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