Identity for generalized Bernoulli polynomials (original) (raw)

Some Identities of Symmetry for the Generalized Bernoulli Numbers and Polynomials

Abstract and Applied Analysis, 2009

By the properties ofp-adic invariant integral onℤp, we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties ofp-adic invariant integral onℤp, we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.

Some new identities on the Apostol-Bernoulli polynomials of higher order derived from Bernoulli basis

2013

In the present paper, we aim to obtain some new interesting relations and identities of the Apostol-Bernoulli polynomials of higher order. These identities are derived using a suitable polynomial basis, for which we employ a Bernoulli basis. Finally, by utilizing our method, we also derive formulas for the convolutions of Bernoulli and Euler polynomials, expressed via Apostol-Bernoulli polynomials of higher order.

A generalization of the Bernoulli polynomials

Journal of Applied Mathematics, 2003

A generalization of the Bernoulli polynomials and, consequently, of the Bernoulli numbers, is defined starting from suitable generating functions. Furthermore, the differential equations of these new classes of polynomials are derived by means of the factorization method introduced by .

New family of Bernoulli-type polynomials and some application

In this paper, we present a new family of generalized Bernoulli-type polynomials, as well as its numbers. In addition, we obtain some results such as algebraic and differential properties for this new family of Bernoulli-type polynomials. Likewise, the generalized Bernoulli-type polynomials matrix R (α) (x) is introduced. We deduce some product formulae for R (α) (x) and also, the inverse of the Bernoullitype matrix R is determined. Furthermore, we establish some explicit expressions for the Bernoullitype polynomial matrix R(x), which involve the generalized Pascal matrix and finally we study the summation formula of Euler-Maclaurin type and the Riemann zeta function applied to these Bernoullitype polynomials.

Old and New Identities for Bernoulli Polynomials via Fourier Series

International Journal of Mathematics and Mathematical Sciences, 2012

The Bernoulli polynomials B k restricted to [0, 1) and extended by periodicity have nth sine and cosine Fourier coefficients of the form C k /n k . In general, the Fourier coefficients of any polynomial restricted to [0, 1) are linear combinations of terms of the form 1/n k . If we can make this linear combination explicit for a specific family of polynomials then by uniqueness of Fourier series, we get a relation between the given family and the Bernoulli polynomials.

A symbolic approach to some identities for Bernoulli–Barnes polynomials

International Journal of Number Theory, 2015

The Bernoulli–Barnes polynomials are defined as a natural multidimensional extension of the classical Bernoulli polynomials. Many of the properties of the Bernoulli polynomials admit extensions to this new family. A specific expression involving the Bernoulli–Barnes polynomials has recently appeared in the context of self-dual sequences. The work presented here provides a proof of this self-duality using the symbolic calculus attached to Bernoulli numbers and polynomials. Several properties of the Bernoulli–Barnes polynomials are established by this procedure.

Some Identities on Type 2 Degenerate Bernoulli Polynomials of the Second Kind

Symmetry, 2020

In recent years, many mathematicians studied various degenerate versions of some special polynomials for which quite a few interesting results were discovered. In this paper, we introduce the type 2 degenerate Bernoulli polynomials of the second kind and their higher-order analogues, and study some identities and expressions for these polynomials. Specifically, we obtain a relation between the type 2 degenerate Bernoulli polynomials of the second and the degenerate Bernoulli polynomials of the second, an identity involving higher-order analogues of those polynomials and the degenerate Stirling numbers of the second kind, and an expression of higher-order analogues of those polynomials in terms of the higher-order type 2 degenerate Bernoulli polynomials and the degenerate Stirling numbers of the first kind.

On Gessel-Kaneko's identity for Bernoulli numbers

Applicable Analysis and Discrete Mathematics, 2013

The present work deals with Bernoulli numbers. Using Zeilberger's algorithm, we generalize an identity on Bernoulli numbers of Gessel-Kaneko's type. Appendix written by Ira M. Gessel offers a closely related formula via umbral calculus.

A new class of generalized polynomials associated with Laguerre and Bernoulli polynomials

TURKISH JOURNAL OF MATHEMATICS

Motivated by their importance and potential for applications in certain problems in number theory, combinatorics, classical and numerical analysis, and other fields of applied mathematics, a variety of polynomials and numbers with their variants and extensions have recently been introduced and investigated. In this paper, we aim to introduce generalized Laguerre-Bernoulli polynomials and investigate some of their properties such as explicit summation formulas, addition formulas, implicit formulas, and symmetry identities. Relevant connections of the results presented here with those relatively simple numbers and polynomials are considered.