*Review* **Convolutions for Bernoulli and Euler–Genocchi Polynomials of Order (***<sup>r</sup>***,***<sup>m</sup>***) and Their Probabilistic Interpretation †**

**Robert Frontczak 1,‡ and Živorad Tomovski 2,\***


**Abstract:** The main purpose of this article is to derive several convolutions for generalized Bernoulli and Euler–Genocchi polynomials of order (*r*, *<sup>m</sup>*), *<sup>B</sup>*(*<sup>r</sup>*,*<sup>m</sup>*) *n* (*x*) and *<sup>A</sup>*(*<sup>r</sup>*,*<sup>m</sup>*) *n* (*x*), respectively. These polynomials have been introduced recently and contain the generalized Bernoulli, Euler and Genocchi polynomials as special members. Some of our results extend the results of M. Merca and others concerning Bernoulli numbers and polynomials. Probabilistic interpretations of the presented results are also given.

**Keywords:** Bernoulli number; generalized Bernoulli polynomial; generalized Euler–Genocchi polynomial; functional equation; convolution; probability distribution; moment-generating function; moments

**MSC:** 11B68; 11S40; 05A15; 60E05
