On the supercongruence conjectures of van Hamme
On the supercongruence conjectures of van Hamme
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DOI：
10.1186/s40687-015-0037-6
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发表时间：
2015-12-01
影响因子：
1.2
通讯作者：
Swisher, Holly
中科院分区：
文献类型：
--
作者：
Swisher, Holly
1997年，van Hamme发展了素数p的p-进模拟，这是由Ramanujan最初研究的将超几何级数与Gamma函数值联系起来的几个级数。这些类比将超几何级数的截断和与p-进Gamma函数的值联系起来，称为Ramanujan型超同余。范哈姆总共猜想了13个这样的公式，其中3个是由范哈姆自己证明的，另外5个是最近用广泛的方法证明的。在这里，我们研究剩下的五个van Hamme超同余中的四个，重温一些已证明的超同余，并提供一些扩展。
In 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.