Some supercongruences occurring in truncated hypergeometric series
Some supercongruences occurring in truncated hypergeometric series
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DOI：
10.1016/j.aim.2015.11.043
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发表时间：
2016-02-26
影响因子：
1.7
通讯作者：
Ramakrishna, Ravi
中科院分区：
文献类型：
--
作者：
Long, Ling;Ramakrishna, Ravi
超同余是指终止超几何级数和p-adic Gamma函数的同元之间的同余，它们比人们期望用交换形式群律证明的要强。我们利用经典的超几何变换公式证明了一些这样的超同余。这些公式，其中大多数是几十年或几百年前的，允许我们将终止序列写成伽马值乘积的比率。在这一点上，总和已经成为一种共识。将这些Gamma-同余式写成Gamma（p）-同余式，我们就处于一种非常适合证明p-adic同余式的情形。这些Gamma（p）-函数可以用它们的泰勒级数展开式p-逼近。有时会有低阶项的抵消，导致更强的同余。使用这种技术，我们证明，除其他事项外，猜想Kibelbek和加强版本的猜想货车Hamme。(C)2015 Elsevier Inc. All rights reserved.
For the purposes of this paper supercongruences are congruences between terminating hypergeometric series and quotients of p-adic Gamma functions that are stronger than those one can expect to prove using commutative formal group laws. We prove a number of such supercongruences by using classical hypergeometric transformation formulae. These formulae, most of which are decades or centuries old, allow us to write the terminating series as the ratio of products of Gamma-values. At this point sums have become quotients. Writing these Gamma-quotients as Gamma(p)-quotients, we are in a situation that is well suited for proving p-adic congruences. These Gamma(p)-functions can be p-adically approximated by their Taylor series expansions. Sometimes there is cancellation of the lower order terms, leading to stronger congruences. Using this technique we prove, among other things, a conjecture of Kibelbek and a strengthened version of a conjecture of van Hamme. (C) 2015 Elsevier Inc. All rights reserved.