Identities involving Bernoulli and Euler polynomials

Abstract

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