关于van Hamme的超同余
On the supercongruence conjectures of van Hamme
摘要：
在1997年，范·哈姆为素数$p$开发了$p-$进类比，它将超几何级数与最初由拉马努金研究的Γ函数的值联系起来。这些类似物将超几何级数的截断和与$p-$ adic Γ函数的值联系起来，称为Ramanujan型超同余。在所有，凡Hamme estretured 13这样的公式，其中三个是证明了凡Hamme自己，和其他五个已证明最近使用广泛的方法。在这里，我们将探讨剩下的五个van Hamme超同余中的四个，重新审视一些已证明的超同余，并提供一些扩展。
英文摘要：
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.
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