On the zeros of shifted Bernoulli and Euler polynomials

In this short survey paper, some recent results on the zero-structure of shifted Bernoulli polynomials $B_k(X)+b$ and Euler polynomials $E_k(X)+b$ are presented. Further, using some previous results, we show that there are only finitely many pairs $(k,b)$ with $k\geq 3$, $b\in \mathbb{C}$ for which $B_k(X)+b$, resp., $E_k(X)+b$ has no three simple zeros, and we give explicitly all these pairs $(k,b)$.