Rota-Baxter operators and Bernoulli polynomials

Vsevolod Gubarev

doi:10.2478/cm-2021-0001

Rota-Baxter operators and Bernoulli polynomials

Communications in Mathematics ◽

10.2478/cm-2021-0001 ◽

2021 ◽

Vol 29 (1) ◽

pp. 1-14

Author(s):

Vsevolod Gubarev

Keyword(s):

Bernoulli Polynomials ◽

Mathematical Physics ◽

Power Sum

Abstract We develop the connection between Rota-Baxter operators arisen from algebra and mathematical physics and Bernoulli polynomials. We state that a trivial property of Rota-Baxter operators implies the symmetry of the power sum polynomials and Bernoulli polynomials. We show how Rota-Baxter operators equalities rewritten in terms of Bernoulli polynomials generate identities for the latter.

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Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials

Symmetry ◽

10.3390/sym11070847 ◽

2019 ◽

Vol 11 (7) ◽

pp. 847 ◽

Cited By ~ 4

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.

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On the Symmetries of theq-Bernoulli Polynomials

Abstract and Applied Analysis ◽

10.1155/2008/914367 ◽

2008 ◽

Vol 2008 ◽

pp. 1-7 ◽

Cited By ~ 4

Author(s):

Taekyun Kim

Keyword(s):

Bernoulli Polynomials ◽

Bernoulli Numbers ◽

Reflection Symmetry ◽

Bernoulli Numbers And Polynomials ◽

Power Sum ◽

Congruence Properties

Kupershmidt and Tuenter have introduced reflection symmetries for theq-Bernoulli numbers and the Bernoulli polynomials in (2005), (2001), respectively. However, they have not dealt with congruence properties for these numbers entirely. Kupershmidt gave a quantization of the reflection symmetry for the classical Bernoulli polynomials. Tuenter derived a symmetry of power sum polynomials and the classical Bernoulli numbers. In this paper, we study the new symmetries of theq-Bernoulli numbers and polynomials, which are different from Kupershmidt's and Tuenter's results. By using our symmetries for theq-Bernoulli polynomials, we can obtain some interesting relationships betweenq-Bernoulli numbers and polynomials.

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Sums of Products Involving Power Sums of φ(n) Integers

Journal of Numbers ◽

10.1155/2014/158351 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6

Author(s):

Jitender Singh

Keyword(s):

Closed Form ◽

Complex Variable ◽

Bernoulli Polynomials ◽

Rational Numbers ◽

Bernoulli Numbers ◽

Higher Order ◽

Power Sums ◽

Power Sum ◽

Mobius Function ◽

Sums Of Products

A sequence of rational numbers as a generalization of the sequence of Bernoulli numbers is introduced. Sums of products involving the terms of this generalized sequence are then obtained using an application of Faà di Bruno's formula. These sums of products are analogous to the higher order Bernoulli numbers and are used to develop the closed form expressions for the sums of products involving the power sums Ψk(x,n):=∑d|n‍μ(d)dkSkx/d, n∈ℤ+ which are defined via the Möbius function μ and the usual power sum Sk(x) of a real or complex variable x. The power sum Sk(x) is expressible in terms of the well-known Bernoulli polynomials by Sk(x):=(Bk+1(x+1)-Bk+1(1))/(k+1).

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