Recurrences for Bernoulli and Euler polynomials and numbers (Q1573562)

This article contains both classical and new results on the subject of the title. One starting point is the following nice relation for Bernoulli polynomials \(B_m(X)\), where \(X\) and \(Y\) are indeterminates and \(m\) and \(l\) are positive integers: \[ \left(B_m(X)+Y\right)^l = \left(B_l(X+Y)-Y\right)^m \] (in the usual symbolic notation). The corresponding relation also holds for Euler polynomials \(E_m(X)\). Using these formulas, the author proves a series of other recurrences for \(B_m(X)\) and \(E_m(X)\) as well as for Bernoulli and Euler numbers. The proofs are often simpler and shorter than the previously known proofs.