Congruences involving $g_n(x)=\sum_{k=0}^n\binom nk^2\binom{2k}kx^k$

Zhi-Wei Sun

paper

Congruences involving $g_{n} (x) = \sum_{k = 0}^{n} (k n)^{2} (k 2 k) x^{k}$

arXiv:1407.0967

Abstract

Define $g_{n} (x) = \sum_{k = 0}^{n} (k n)^{2} (k 2 k) x^{k}$ 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 $k = 1 \sum p - 1 \frac{g _{k} ( - 1 )}{k} \equiv 0 (mod p^{2}) an d k = 1 \sum p - 1 \frac{g _{k} ( - 1 )}{k ^{2}} \equiv 0 (mod p) .$ This is similar to Wolstenholme's classical congruences $k = 1 \sum p - 1 \frac{1}{k} \equiv 0 (mod p^{2}) an d k = 1 \sum p - 1 \frac{1}{k ^{2}} \equiv 0 (mod p)$ for any prime $p > 3$ .

22 pages, final published version

Cited by in corpus (1)

Two congruences involving harmonic numbers with applications

Congruences involving gn​(x)=∑k=0n​(kn​)2(k2k​)xk

Abstract

Cited by in corpus (1)

Congruences involving $g_{n} (x) = \sum_{k = 0}^{n} (k n)^{2} (k 2 k) x^{k}$