Congruences Involving G_n(x)=\sum_{k=0}^n\binom Nk^2 ... - ArXiv

Mathematics > Number Theory arXiv:1407.0967 (math) [Submitted on 3 Jul 2014 (v1), last revised 19 Jul 2016 (this version, v8)] Title:Congruences involving $g_n(x)=\sum_{k=0}^n\binom nk^2\binom{2k}kx^k$ Authors:Zhi-Wei Sun View a PDF of the paper titled Congruences involving $g_n(x)=\sum_{k=0}^n\binom nk^2\binom{2k}kx^k$, by Zhi-Wei Sun View PDF

Abstract:Define $g_n(x)=\sum_{k=0}^n\binom nk^2\binom{2k}kx^k$ for $n=0,1,2,...$. Those numbers $g_n=g_n(1)$ are closely related to Apéry numbers and Franel numbers. In this paper we establish some fundamental congruences involving $g_n(x)$. For example, for any prime $p>5$ we have $$\sum_{k=1}^{p-1}\frac{g_k(-1)}{k}\equiv 0\pmod{p^2}\quad{and}\quad\sum_{k=1}^{p-1}\frac{g_k(-1)}{k^2}\equiv 0\pmod p.$$ This is similar to Wolstenholme's classical congruences $$\sum_{k=1}^{p-1}\frac1k\equiv0\pmod{p^2}\quad{and}\quad\sum_{k=1}^{p-1}\frac{1}{k^2}\equiv0\pmod p$$ for any prime $p>3$.

Comments:	22 pages, final published version
Subjects:	Number Theory (math.NT); Combinatorics (math.CO)
MSC classes:	11A07, 11B65, 05A10, 05A30, 11B75
Cite as:	arXiv:1407.0967 [math.NT]
(or arXiv:1407.0967v8 [math.NT] for this version)
https://doi.org/10.48550/arXiv.1407.0967 Focus to learn more arXiv-issued DOI via DataCite
Journal reference:	Ramanujan J. 40(2016), no.3, 511-533

Submission history

From: Zhi-Wei Sun [view email] [v1] Thu, 3 Jul 2014 15:58:17 UTC (9 KB) [v2] Tue, 8 Jul 2014 16:04:17 UTC (9 KB) [v3] Thu, 10 Jul 2014 16:59:36 UTC (10 KB) [v4] Wed, 23 Jul 2014 19:58:50 UTC (11 KB) [v5] Tue, 23 Dec 2014 17:58:39 UTC (10 KB) [v6] Mon, 29 Dec 2014 11:59:26 UTC (10 KB) [v7] Tue, 20 Oct 2015 15:42:31 UTC (10 KB) [v8] Tue, 19 Jul 2016 15:51:14 UTC (10 KB) Full-text links:

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