New super congruences involving Bernoulli and Euler polynomials

TL;DR

This paper derives new super congruences involving sums of binomial coefficients and rational p-adic integers, expressed through Bernoulli and Euler polynomials, and provides transformation formulas for these congruences modulo p^2.

Contribution

It establishes novel super congruences for specific binomial coefficient sums in terms of Bernoulli and Euler polynomials, including transformation formulas modulo p^2.

Findings

Congruences for sums involving binomial coefficients and p-adic integers in terms of Bernoulli and Euler polynomials.

Transformation formulas for these congruences modulo p^2.

Results applicable for primes greater than 3.

Abstract

Let $p>3$ be a prime, and let $a$ be a rational p-adic integer with $a\not\equiv 0\pmod p$. In this paper we establish congruences for $$\sum_{k=1}^{(p-1)/2}\frac{\binom ak\binom{-1-a}k}k, \quad\sum_{k=0}^{(p-1)/2}k\binom ak\binom{-1-a}k \quad\text{and}\quad\sum_{k=0}^{(p-1)/2}\frac{\binom ak\binom{-1-a}k}{2k-1}\pmod {p^2}$$ in terms of Bernoulli and Euler polynomials. We also give some transformation formulas for congruences modulo $p^2$.