On p-Bernoulli numbers and polynomials

In this paper we define a new family of p-Bernoulli numbers, which are derived from the Gaussian hypergeometric function, and we establish some basic properties. Based on a three-term recurrence relation, an algorithm for computing Bernoulli numbers is given. A similar algorithm for Bernoulli polynomials is also presented. We show that the p-Bernoulli numbers are related to the certain regular values of the Euler–Zagier's multiple zeta function at non-positive integers of depth p. Finally, a generalization and some applications on m-Fubini numbers are given.