Symmetric identities for the generalized higher-order q-Bernoulli polynomials

Some Symmetric Identities Involving the Stirling Polynomials Under the Finite Symmetric Group

Mathematics ◽

10.3390/math6120332 ◽

2018 ◽

Vol 6 (12) ◽

pp. 332

Author(s):

Dongkyu Lim ◽

Feng Qi

Keyword(s):

Symmetric Group ◽

Bernoulli Polynomials ◽

Higher Order ◽

Finite Symmetric Group ◽

Symmetric Identities

In the paper, the authors present some symmetric identities involving the Stirling polynomials and higher order Bernoulli polynomials under all permutations in the finite symmetric group of degree n. These identities extend and generalize some known results.

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

Symmetry ◽

10.3390/sym13040648 ◽

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.

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Two-Variable Type 2 Poly-Fubini Polynomials

Mathematics ◽

10.3390/math9030281 ◽

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.

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Symmetric Identities of Hermite-Bernoulli Polynomials and Hermite-Bernoulli Numbers Attached to a Dirichlet Character χ

Symmetry ◽

10.3390/sym10120675 ◽

2018 ◽

Vol 10 (12) ◽

pp. 675 ◽

Cited By ~ 1

Author(s):

Serkan Araci ◽

Waseem Khan ◽

Kottakkaran Nisar

Keyword(s):

Bernoulli Polynomials ◽

Dirichlet Character ◽

Bernoulli Numbers ◽

Complex Order ◽

Arbitrary Complex ◽

Symmetric Identities

We aim to introduce arbitrary complex order Hermite-Bernoulli polynomials and Hermite-Bernoulli numbers attached to a Dirichlet character χ and investigate certain symmetric identities involving the polynomials, by mainly using the theory of p-adic integral on Z p . The results presented here, being very general, are shown to reduce to yield symmetric identities for many relatively simple polynomials and numbers and some corresponding known symmetric identities.

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Irreducibility of Bernoulli Polynomials of Higher Order

Canadian Journal of Mathematics ◽

10.4153/cjm-1962-047-2 ◽

1962 ◽

Vol 14 ◽

pp. 565-567 ◽

Cited By ~ 1

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.

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Some Symmetric Identities for the Multiple (p, q)-Hurwitz-Euler eta Function

Symmetry ◽

10.3390/sym11050645 ◽

2019 ◽

Vol 11 (5) ◽

pp. 645 ◽

Cited By ~ 4

Author(s):

Kyung-Won Hwang ◽

Cheon Seoung Ryoo

Keyword(s):

Higher Order ◽

Complex Field ◽

Euler Numbers ◽

Euler Numbers And Polynomials ◽

Symmetric Identities ◽

Symmetric Properties

The main purpose of this paper is to find some interesting symmetric identities for the ( p , q ) -Hurwitz-Euler eta function in a complex field. Firstly, we define the multiple ( p , q ) -Hurwitz-Euler eta function by generalizing the Carlitz’s form ( p , q ) -Euler numbers and polynomials. We find some formulas and properties involved in Carlitz’s form ( p , q ) -Euler numbers and polynomials with higher order. We find new symmetric identities for multiple ( p , q ) -Hurwitz-Euler eta functions. We also obtain symmetric identities for Carlitz’s form ( p , q ) -Euler numbers and polynomials with higher order by using symmetry about multiple ( p , q ) -Hurwitz-Euler eta functions. Finally, we study the distribution and symmetric properties of the zero of Carlitz’s form ( p , q ) -Euler numbers and polynomials with higher order.

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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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