A Family of the r-Associated Stirling Numbers of the Second Kind and Generalized Bernoulli Polynomials

In this article, we derive representation formulas for a class of r-associated Stirling numbers of the second kind and examine their connections with a class of generalized Bernoulli polynomials. Herein, we use the Blissard umbral approach and the familiar Bell polynomials. Links with available literature on this subject are also pointed out. The extension to the bivariate case is discussed.

Hankel determinants of sequences related to Bernoulli and Euler polynomials

We evaluate the Hankel determinants of various sequences related to Bernoulli and Euler numbers and special values of the corresponding polynomials. Some of these results arise as special cases of Hankel determinants of certain sums and differences of Bernoulli and Euler polynomials, while others are consequences of a method that uses the derivatives of Bernoulli and Euler polynomials. We also obtain Hankel determinants for sequences of sums and differences of powers and for generalized Bernoulli polynomials belonging to certain Dirichlet characters with small conductors. Finally, we collect and organize Hankel determinant identities for numerous sequences, both new and known, containing Bernoulli and Euler numbers and polynomials.

Symmetric identity for polynomial sequences satisfying A′ n +1(x) = (n + 1)An (x)

Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying A′ n +1(x) = (n + 1)An (x) with A 0(x) a constant polynomial. This identity allows us to obtain in a simple way some known relations involving Apostol-Bernoulli polynomials, Apostol-Euler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character.

A three-term recurrence formula for the generalized Bernoulli polynomials

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 results on the q-generalized Bernoulli polynomials of level m

This paper aims to show new algebraic properties from the q-generalized Bernoulli polynomials B_n^{[m - 1]}(x;q) of level m, as well as some others identities which connect this polynomial class with the q-generalized Bernoulli polynomials of level m, as well as the q-gamma function, and the q-Stirling numbers of the second kind and the q-Bernstein polynomials.

The actions on the generating functions for the family of the generalized Bernoulli polynomials

In this paper, we study the generalization Bernoulli numbers and polynomials attached to a periodic group homomorphism from a finite cyclic group to the set of complex numbers and derive new periodic group homomorphism by using a fixed periodic group homomorphism. Hence, we obtain not only multiplication formulas, but also some new identities for the generalized Bernoulli polynomials attached to a periodic group homomorphism.