A three-term recurrence formula for the generalized Bernoulli polynomials

Abstract

In the present paper, we propose some new explicit formulas of the higher order Daehee polynomials in terms of the generalized r-Stirling and r-Whitney numbers of the second kind. As a consequence, we derive a three-term recurrence formula for the calculation of the generalized Bernoulli polynomials of order k.

New recurrence relations for several classical families of polynomials

“…In this paper, we consider the following families of polynomials: E α m (x), Bernoulli polynomials of the second kind b α m (x), Buchholz polynomials P a m (x), generalized Bessel polynomials Y a m (x) and generalized Apostol-Euler polynomials E α m (x; λ). Several (more or less involved) explicit formulas, recurrence relations and properties of the generalized Bernoulli polynomials may be found in [3,4,6,8,19,22,23,28,29,31]; of the generalized Euler polynomials in [6,8,12,29,31]; of the Bernoulli polynomials of the second kind in [9,25,30]; of the Buchholz polynomials in [1,2,7,18]; of the generalized Bessel polynomials in [10,13,16,24]; and of the generalized Apostol-Euler polynomials in [5,14,20,21,27,32]. See also references therein for further information.…”