On Well-Poised Series, Generalized Hypergeometric Series having Parameters in Pairs, each Pair with the Same Sum

Whipple, F. J. W.

doi:10.1112/plms/s2-24.1.247

Cited by 60 publications

(38 citation statements)

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“…(bp) k ) ≥ 4, however the k = 5 case remains elusive. We thus obtain that for primes p > 5, (15) Γ p (a + bp) For this section we will make use of the following identities of Whipple (see (5.1) and (6.3) in [17])…”

Section: 2mentioning

confidence: 99%

On the supercongruence conjectures of van Hamme

Swisher

2015

Res Math Sci

129

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In 1914, Ramanujan listed 17 infinite series representations of 1/π, including for example Several of Ramanujan's formulas relate hypergeometric series to values of the gamma function.In the 1980s, it was discovered that Ramanujan's formulas provided efficient means for calculating digits of π. In 1987, Borwein and Borwein [3] proved all 17 of Ramanujan's identities, while Chudnovsky and Chudnovsky [5] derived additional series for 1/π. Digits of π were calculated in both papers resulting in a new world record at the time by the Chudnovskys of 2, 260, 331, 336 digits. All of these Ramanujan-type formulas for 1/π are related to elliptic curves with complex multiplication (CM).In 1997, van Hamme [18] developed p-adic analogs, for primes p, of several Ramanujan-type series. Analogs of this type are called Ramanujan-type supercongruences, and relate truncated sums of hypergeometric series to values of the p-adic gamma function. In a recent paper [4], the author along with S. Chisholm, A. Deines, L. Long, and G. Nebe proves a general p-adic analog of Ramanujan-type supercongruences modulo p 2 AbstractIn 1997, van Hamme developed p-adic analogs, for primes p, of several series which relate hypergeometric series to values of the gamma function, originally studied by Ramanujan. These analogs relate truncated sums of hypergeometric series to values of the p-adic gamma function, and are called Ramanujan-type supercongruences. In all, van Hamme conjectured 13 such formulas, three of which were proved by van Hamme himself, and five others have been proved recently using a wide range of methods.Here, we explore four of the remaining five van Hamme supercongruences, revisit some of the proved ones, and provide some extensions.

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Section: 2mentioning

confidence: 99%

On the supercongruence conjectures of van Hamme

Swisher

2015

Res Math Sci

129

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“…Форму-ла (44) справедлива при |z| < 1, а также при любом z ∈ C, если a является неположительным целым. В работе [148] Ф. Уиппл назвал вполне уравновешенными гипергеометриче-ские ряды, удовлетворяющие условию…”

Section: в в зудилинunclassified

Арифметические Гипергеометрические Ряды

Zudilin¹

2011

Uspekhi Mat. Nauk

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“…Note that the series terminates if some a j is a negative integer. In [30], Whipple studied properties of well-poised series where p = q + 1, z = ±1, and a 1 + 1 = a 2 + b 1 = a 3 + b 2 = · · · = a p + b q . One such transformation property of the well-poised series (see (6.3), page 252 in [30]) is…”

Section: Preliminariesmentioning

confidence: 99%