scholarly journals On the Identities of Symmetry for the Generalized Bernoulli Polynomials Attached to of Higher Order

2009 ◽  
Vol 2009 (1) ◽  
pp. 640152 ◽  
Author(s):  
Taekyun Kim ◽  
Lee-Chae Jang ◽  
Young-Hee Kim ◽  
Kyung-Won Hwang
Keyword(s):  
Bernoulli Polynomials ◽  
Higher Order ◽  
Generalized Bernoulli Polynomials

scholarly journals A three-term recurrence formula for the generalized Bernoulli polynomials

2021 ◽  
Vol 39 (6) ◽  
pp. 139-145
Author(s):  
Mohamed Amine Boutiche ◽  
Ghania Guettai ◽  
Mourad Rahmani ◽  
Madjid Sebaoui
Keyword(s):  
Recurrence Formula ◽  
Bernoulli Polynomials ◽  
Higher Order ◽  
Explicit Formulas ◽  
Whitney Numbers ◽  
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.

scholarly journals A New Class of Higher-Order Hypergeometric Bernoulli Polynomials Associated with Lagrange–Hermite Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 648
Author(s):  
Ghulam Muhiuddin ◽  
Waseem Ahmad Khan ◽  
Ugur Duran ◽  
Deena Al-Kadi
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Laguerre Polynomials ◽  
Hermite Polynomials ◽  
Bernoulli Polynomials ◽  
Higher Order ◽  
New Class

The purpose of this paper is to construct a unified generating function involving the families of the higher-order hypergeometric Bernoulli polynomials and Lagrange–Hermite polynomials. Using the generating function and their functional equations, we investigate some properties of these polynomials. Moreover, we derive several connected formulas and relations including the Miller–Lee polynomials, the Laguerre polynomials, and the Lagrange Hermite–Miller–Lee polynomials.

scholarly journals Two-Variable Type 2 Poly-Fubini Polynomials

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 281
Author(s):  
Ghulam Muhiuddin ◽  
Waseem Ahmad Khan ◽  
Ugur Duran
Keyword(s):  
Integral Representation ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Higher Order ◽  
Variable Type ◽  
Summation Formulas

In the present work, a new extension of the two-variable Fubini polynomials is introduced by means of the polyexponential function, which is called the two-variable type 2 poly-Fubini polynomials. Then, some useful relations including the Stirling numbers of the second and the first kinds, the usual Fubini polynomials, and the higher-order Bernoulli polynomials are derived. Also, some summation formulas and an integral representation for type 2 poly-Fubini polynomials are investigated. Moreover, two-variable unipoly-Fubini polynomials are introduced utilizing the unipoly function, and diverse properties involving integral and derivative properties are attained. Furthermore, some relationships covering the two-variable unipoly-Fubini polynomials, the Stirling numbers of the second and the first kinds, and the Daehee polynomials are acquired.

scholarly journals Some Identities of Symmetry for the Generalized Bernoulli Numbers and Polynomials

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Taekyun Kim ◽  
Seog-Hoon Rim ◽  
Byungje Lee
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sums ◽  
Invariant Integral ◽  
Generalized Bernoulli Polynomials ◽  
Symmetric Properties

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.

Irreducibility of Bernoulli Polynomials of Higher Order

1962 ◽  
Vol 14 ◽  
pp. 565-567 ◽  
Author(s):  
P. J. McCarthy
Keyword(s):  
Positive Integer ◽  
Bernoulli Number ◽  
Bernoulli Polynomials ◽  
Constant Term ◽  
Rational Numbers ◽  
Higher Order ◽  
Image Position

The Bernoulli polynomials of order k, where k is a positive integer, are defined byBm(k)(x) is a polynomial of degree m with rational coefficients, and the constant term of Bm(k)(x) is the mth Bernoulli number of order k, Bm(k). In a previous paper (3) we obtained some conditions, in terms of k and m, which imply that Bm(k)(x) is irreducible (all references to irreducibility will be with respect to the field of rational numbers). In particular, we obtained the following two results.

scholarly journals Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 847 ◽  
Author(s):  
Dmitry V. Dolgy ◽  
Dae San Kim ◽  
Jongkyum Kwon ◽  
Taekyun Kim
Keyword(s):  
Random Variables ◽  
Infinite Family ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Integer Power ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sums ◽  
Power Sum ◽  
Invariant Integrals

In this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain p-adic invariant integrals on Z p . In particular, we derive various expressions for the polynomials associated with integer power sums, called integer power sum polynomials and also for their degenerate versions. Further, we compute the expectations of an infinite family of random variables which involve the degenerate Stirling polynomials of the second and some value of higher-order Bernoulli polynomials.

scholarly journals New results on the q-generalized Bernoulli polynomials of level m

2019 ◽  
Vol 52 (1) ◽  
pp. 511-522
Author(s):  
Alejandro Urieles ◽  
María José Ortega ◽  
William Ramírez ◽  
Samuel Vega
Keyword(s):  
Gamma Function ◽  
Bernstein Polynomials ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Algebraic Properties ◽  
Generalized Bernoulli Polynomials ◽  
Polynomial Class

AbstractThis 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.

Some identities of symmetry for the generalized higher-order (h,q)-Bernoulli polynomials of the second kind

2014 ◽  
Vol 8 ◽  
pp. 1061-1070
Author(s):  
Dae San Kim ◽  
Taekyun Kim ◽  
Seog-Hoon Rim ◽  
Jong Jin Seo
Keyword(s):  
Bernoulli Polynomials ◽  
Higher Order
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