Probabilistic proofs of some relationships between the Bernoulli and Euler polynomials

Abstract

The main purpose of this article is to provide probabilistic proofs of the relationships between the generalized Bernoulli (or Nörlund) polynomials $B^{alpha}_n (x)$ and the generalized Euler polynomials $B^{alpha}_n (x)$ of (real or complex) orderand degree n in x, which were proved recently by Srivastava and Pintér [11]. Some other approaches to these relationships and their seemingly interesting generalizations are also investigated.