Shortened recurrence relations for generalized Bernoulli numbers and polynomials

Abstract

It is the main purpose of this paper to study shortened recurrence relations for generalized Bernoulli numbers and polynomials attached to χ, χ being a primitive Dirichlet character, in which some of the preceding numbers or polynomials are completely excluded. As a result, we are able to establish several kinds of such type recurrences by generalizing some known identities on classical Bernoulli numbers and polynomials such as Saalschütz–Gelfand and von Ettingshausen–Stern's formulas. Furthermore, we discuss shortened recurrence relations for special values of the Riemann zeta and its allied functions.