A generalization of the Bernoulli polynomials

2013

In this paper we use probabilistic methods to derive some results on the generalized Bernoulli and generalized Euler polynomials. Our approach is based on the properties of Appell polynomials associated with uniformly distributed and Bernoulli distributed random variables and their sums.

International Journal of Mathematics and Mathematical Sciences, 2003

Filomat, 2017

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.

Six approaches to the theory of Bernoulli polynomials are known; these are associated with the names of J. Bernoulli [2], L. Euler [4], E. Lucas [8], P. E. Appell [1], A. Hürwitz [6] and D. H. Lehmer [7]. In this note we deal with a new determinantal definition for Bernoulli polynomials recently proposed by F. Costabile [3]; in particular, we emphasize some consequent procedures for automatic calculation and recover the better known properties of these polynomials from this new definition. Finally, after we have observed the equivalence of all considered approaches, we conclude with a circular theorem that emphasizes the direct equivalence of three of previous approaches.

SN Applied Sciences

In this research paper, the authors derived the non-linear differential equations for certain hybrid special polynomials related to the Bernoulli polynomials. The families of non-linear differential equations arising from the generating functions of the Bernoulli-Euler and Bernoulli-Genocchi polynomials are derived. Further, these non-linear differential equations are used to derive certain identities and formulas for the Bernoulli-Euler and Bernoulli-Genocchi numbers. However, to provided an exception, a linear differential equation is derived from the generating function of the Genocchi-Euler polynomials.

The Journal of Analysis

Motivated by certain problems connected with the stochastic analysis of the recursively defined time series, in this paper, we define and study some polynomial sequences. Beside computation of these polynomials and their connection to the Euler-Apostol numbers, we provesome basic properties and give an interesting connection of these polynomials with the well-known Bernoulli numbers, as well as some new summation formulas for Bernoulli's numbers. Finally, we prove that zeros of these polynomials are simple, real and symmetrically distributed in [0,1].

2016

In this paper, we introduce the degenerate poly-Bernoulli polynomials of the second kind, which reduce in the limit to the poly-Bernoulli polynomials of the second kind. We present several explicit formulas and recurrence relations for these polynomials. Also, we establish a connection between our polynomials and several known families of polynomials.

AIMS Mathematics, 2021

The main object of this article is to present type 2 degenerate poly-Bernoulli polynomials of the second kind and numbers by arising from modified degenerate polyexponential function and investigate some properties of them. Thereafter, we treat the type 2 degenerate unipoly-Bernoulli polynomials of the second kind via modified degenerate polyexponential function and derive several properties of these polynomials. Furthermore, some new identities and explicit expressions for degenerate unipoly polynomials related to special numbers and polynomials are obtained. In addition, certain related beautiful zeros and graphical representations are displayed with the help of Mathematica.

Abstract and Applied Analysis, 2014

The main purpose of this paper is to introduce and investigate a class ofq-Bernoulli,q-Euler, andq-Genocchi polynomials. Theq-analogues of well-known formulas are derived. In addition, theq-analogue of the Srivastava-Pintér theorem is obtained. Some new identities, involvingq-polynomials, are proved.

On certain properties of three parametric kinds of Apostol-type unified Bernoulli-Euler polynomials, 2025

In this paper, we define the three parametric types of Apostol-type unified Bernoulli-Euler polynomials. We present fundamental properties of these polynomials through the utilization of their generating functions. Furthermore, we derive the partial derivatives of these polynomials. Subsequently, we introduce bivariate polynomials and determine their zeros, graphical representations, and approximation values for specific parameters.