On \(p\)-Bernoulli numbers and polynomials. (English) Zbl 1332.11027
Summary: 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.
MSC:
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 11S80 | Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) |
| 11M32 | Multiple Dirichlet series and zeta functions and multizeta values |
Keywords:
Bernoulli numbers and polynomials; Euler-Zagier’s multiple zeta function; Gaussian hypergeometric function; three-term recurrence relationReferences:
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