Supercongruences involving dual sequences

Zhi-Wei Sun

paper

Supercongruences involving dual sequences

arXiv:1512.00712

Abstract

In this paper we study some sophisticated supercongruences involving dual sequences. For $n = 0, 1, 2, \dots$ define $d_{n} (x) = k = 0 \sum n (k n) (k x) 2^{k}$ and $s_{n} (x) = k = 0 \sum n (k n) (k x) (k x + k) = k = 0 \sum n (k n) (- 1)^{k} (k x) (k - 1 - x) .$ For any odd prime $p$ and $p$ -adic integer $x$ , we determine $\sum_{k = 0}^{p - 1} (\pm 1)^{k} d_{k} (x)^{2}$ and $\sum_{k = 0}^{p - 1} (2 k + 1) d_{k} (x)^{2}$ modulo $p^{2}$ ; for example, we establish the new $p$ -adic congruence $k = 0 \sum p - 1 (- 1)^{k} d_{k} (x)^{2} \equiv (- 1)^{⟨ x ⟩_{p}} (mod p^{2}),$ where $⟨ x ⟩_{p}$ denotes the least nonnegative integer $r$ with $x \equiv r (mod p)$ . For any prime $p > 3$ and $p$ -adic integer $x$ , we determine $\sum_{k = 0}^{p - 1} s_{k} (x)^{2}$ modulo $p^{2}$ (or $p^{3}$ if $x \in {0, \dots, p - 1}$ ), and show that $k = 0 \sum p - 1 (2 k + 1) s_{k} (x)^{2} \equiv 0 (mod p^{2}) .$ We also pose several related conjectures.

35 pages, final published version

Supercongruences involving dual sequences

Abstract

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