Super congruences involving Bernoulli and Euler polynomials

2014/07/02 by Zhi-Hong Sun, Sun, Zhi-Hong · 2 citations
Mathematics · #05A19 #11A07 #11B68 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1407.0636

openalex publication_date 2014/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p>3 be a prime, and let a be a rational p-adic integer. Let \B_n(x)\ and \E_n(x)\ denote the Bernoulli polynomials and Euler polynomials, respectively. In this paper we show that ∑_k=0^p-1\binom ak\binom-1-ak≡ (-1)^{⟨ a⟩_p}+ p²t(t+1)E_p-3(-a)\pmodp³ and for a\not≡ -\frac 12\pmod p, ∑_k=0^p-1\binom ak\binom-1-ak\frac 12k+1≡ (1+2t)/(1+2a) +p²(t(t+1))/(1+2a)B_p-2(-a)\pmodp³, where ⟨ a⟩_p∈\0,1,…,p-1\ satisfying a≡ ⟨ a⟩_p\pmod p and t=(a-⟨ a⟩_p)/p. Taking a=-\frac 13,-\frac 14,-\frac 16 in the above congruences we solve some conjectures of Z.W. Sun. In this paper we also establish congruences for ∑_k=0^p-1k\binom ak\binom-1-ak, ∑_k=0^p-1\binom ak\binom-1-ak\frac 12k-1, ∑_k=1^p-1\frac 1k\binom ak\binom-1-ak\pmodp³ and ∑_k=1^p-1\frac (-1)^kk\binom ak, ∑_k=0^p-1\binom ak(-2)^k\pmodp².

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