Recurrence formulas with gaps for Bernoulli and Euler polynomials

TL;DR

This paper derives recurrence formulas with specific gaps for Bernoulli and Euler polynomials and numbers, expanding the understanding of their summation properties with binomial coefficients for particular divisibility conditions.

Contribution

It introduces new formulas for sums involving Bernoulli and Euler polynomials with divisibility constraints, specifically for cases m=2,3,4, which were not previously established.

Findings

Formulas for sums of Bernoulli and Euler polynomials with divisibility conditions

Recurrence relations involving binomial coefficients and Bernoulli/Euler numbers

Extensions to specific cases m=2,3,4

Abstract

Let $\{B_n\}$, $\{B_n(x)\}$ and $\{E_n(x)\}$ be the Bernoulli numbers, Bernoulli polynomials and Euler polynomials, respectively. In this paper we mainly establish formulas for $\sum_{6\mid k-3}\binom nkB_{n-k}(x)$, $\sum_{6\mid k}\binom nkE_{n-k}(x)$ and $\sum_{6\mid k-3}\binom nkm^kB_{n-k}$ in the cases $m=2,3,4$.