Supercongruences involving dual sequences

Abstract

In this paper we study some sophisticated supercongruences involving dual sequences. For n=0,1,2,… definedn(x)=∑k=0n(nk)(xk)2k andsn(x)=∑k=0n(nk)(xk)(x+kk)=∑k=0n(nk)(−1)k(xk)(−1−xk). For any odd prime p and p-adic integer x, we determine ∑k=0p−1(±1)kdk(x)2 and ∑k=0p−1(2k+1)dk(x)2 modulo p2; for example, we establish the new p-adic congruence∑k=0p−1(−1)kdk(x)2≡(−1)〈x〉p(modp2), where 〈x〉p denotes the least nonnegative integer r with x≡r(modp). For any prime p>3 and p-adic integer x, we determine ∑k=0p−1sk(x)2 modulo p2 (or p3 if x∈{0,…,p−1}), and show that∑k=0p−1(2k+1)sk(x)2≡0(modp2). We also pose several related conjectures.