Identities involving Bernoulli and Euler polynomials

TL;DR

This paper derives new identities involving Bernoulli and Euler polynomials, providing formulas that connect these polynomials and numbers, with potential applications in number theory.

Contribution

The paper introduces novel identities linking Bernoulli and Euler polynomials and numbers, expanding the theoretical framework of these classical mathematical objects.

Findings

Derived identities connecting Bernoulli and Euler polynomials.

Established formulas for Bernoulli and Euler numbers.

Provided applications leading to new number formulas.

Abstract

We present various identities involving the classical Bernoulli and Euler polynomials. Among others, we prove that $$ \sum_{k=0}^{[n/4]}(-1)^k {n\choose 4k}\frac{B_{n-4k}(z) }{2^{6k}} =\frac{1}{2^{n+1}}\sum_{k=0}^{n} (-1)^k \frac{1+i^k}{(1+i)^k} {n\choose k}{B_{n-k}(2z)} $$ and $$ \sum_{k=1}^{n} 2^{2k-1} {2n\choose 2k-1} B_{2k-1}(z) = \sum_{k=1}^n k \, 2^{2k} {2n\choose 2k} E_{2k-1}(z). $$ Applications of our results lead to formulas for Bernoulli and Euler numbers, like, for instance, $$ n E_{n-1} =\sum_{k=1}^{[n/2]} \frac{2^{2k}-1}{k} (2^{2k}-2^n){n\choose 2k-1} B_{2k}B_{n-2k}. $$