# Higher-Order Bernoulli Polynomials

## Top PDF Higher-Order Bernoulli Polynomials: ### A note on higher order Bernoulli polynomials

Kim and Kim Journal of Inequalities and Applications 2013, 2013:111 http://www.journaloﬁnequalitiesandapplications.com/content/2013/1/111.. A note on higher-order Bernoulli polynomials D[r] ### Symmetry Properties of Higher Order Bernoulli Polynomials

Hindawi Publishing Corporation Advances in Difference Equations Volume 2009, Article ID 318639, 6 pages doi 10 1155/2009/318639 Research Article Symmetry Properties of Higher Order Bernoulli Polynomia[.] ### Higher order Bernoulli, Euler and Hermite polynomials

Kim et al Advances in Difference Equations 2013, 2013 103 http //www advancesindifferenceequations com/content/2013/1/103 R E V IE W Open Access Higher order Bernoulli, Euler and Hermite polynomials D[.] ### Some Identities Related with the Higher-order Deformed Degenerate Bernoulli and Euler Polynomials

When = 0 , ℰ : (K) = ℰ : (K, 0) are called the degenerate Euler numbers. Note that lim M→' ℰ : (K, ) = ℰ( ) . For 1 ∈ ℕ , Carlitz [1, 2] studied the higher-order degenerate Bernoulli polynomials and the higher-order degenerate Euler polynomials which are defined by the generating functions, respectively ### On the Distribution of the Euler Polynomials and the Genocchi Polynomials of Higher Order

The purpose of this paper is to give the distribution of extended higher order q-Euler and q-Genocchi polynomials. In 24, Choi-Anderson-Srivastava have studied the q-extension of the Apostol-Euler polynomials of order n, and the multiple Hurwitz zeta functions see 24. Actually, their results and definitions are not new see 18, 20 and the definition of the Apostol-Bernoulli numbers in their paper are exactly the same as the definition of the q-extension of Genocchi numbers. Finally, they conjecture that the following q-distribution relation holds: ### Fourier series of higher order Bernoulli functions and their applications

13. Kim, T, Kim, DS, Kwon, HI: Some identities relating to degenerate Bernoulli polynomials. Filomat 30(4), 905-912 (2016) 14. Kim, T: Note on the Euler numbers and polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 17(2), 131-136 (2008) 15. Kim, T: Euler numbers and polynomials associated with zeta functions. Abstr. Appl. Anal. 2008 Art. ID 581582 (2008) 16. Kim, T, Choi, J, Kim, YH: A note on the values of Euler zeta functions at positive integers. Adv. Stud. Contemp. Math. ### A Unification of the Generalized Multiparameter Apostol-type Bernoulli, Euler, Fubini, and Genocchi Polynomials of Higher Order

In recent years, extensive researches on various families of numbers and polynomials such as the Bernoulli numbers and polynomials, Euler numbers and polynomials, Genocchi numbers and polynomials, Fubini numbers and polynomials, and also their generalizations and unifications (see, for instance the recent works of [1, 3, 4, 10, 11, 17, 25, 26, 28, 31]) have become popular due to the abundance of their applications in many branches of mathematics such as in p-adic analytic number theory, umbral calculus, special functions and mathematical analysis, numerical analysis, combinatorics and other related fields. This motivates the author to obtain and explore a new unification of some of the recent generalizations of these special types of polynomials. ### Some identities of higher order Bernoulli, Euler, and Hermite polynomials arising from umbral calculus

Let us assume that P is the algebra of polynomials in the variable x over C and that P ∗ is the vector space of all linear functionals on P . L | p(x) denotes the action of the linear functional L on a polynomial p(x), and we remind that the vector space structure on P ∗ is deﬁned by ### A Note on Genocchi Polynomials and Numbers of Higher Order

Recently, Kim 1 studied q-Genocchi and Euler numbers using Fermionic q-integral and introduced related applications. Kim 2 also gives the q-extensions of the Euler numbers which can be viewed as interpolating of q-analogue of Euler zeta function at negative integers and gives Bernoulli numbers at negative integers by interpolating Riemann zeta function. These numbers are very useful for number theory and mathematical physics. Kim 3, 4 studied q-Bernoulli numbers and polynomials related to Gaussian binomial coeﬃcient and studied also some identities of q-Euler polynomials and q-stirling numbers. Kim 5 made Dedekind DC sum in the meaning of extension of Dedekind sum or Hardy sum and introduced lots of interesting results. The purpose of this paper is to investigate several arithmetic properties of h, q-Genocchi polynomials and numbers of higher order. ### Bernoulli Basis and the Product of Several Bernoulli Polynomials

Kim, “Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on Zp ,” Russian Journal of Mathematical Physics, vol.. Simsek, “[r] ### A Note on Symmetric Properties of the Twisted Bernoulli Polynomials and the Twisted Generalized Bernoulli Polynomials

We define the twisted q-Bernoulli polynomials and the twisted generalized q-Bernoulli polynomi- als attached to χ of higher order and investigate some symmetric properties of them. Furthermore, using these symmetric properties of them, we can obtain some relationships between twisted q- Bernoulli numbers and polynomials and between twisted generalized q-Bernoulli numbers and polynomials. ### Multiparameter Higher Order Daehee and Bernoulli Numbers and Polynomials

This paper gives a new generalization of higher order Daehee and Bernoulli numbers and polynomials. We define the multiparameter higher order Dae- hee numbers and polynomials of the first and second kind. Moreover, we de- rive some new results for these numbers and polynomials. The relations be- tween these numbers and Stirling and Bernoulli numbers are obtained. Fur- thermore, some interesting special cases of the generalized higher order Dae- hee and Bernoulli numbers and polynomials are deduced. ### Series of sums of products of higher order Bernoulli functions

cial polynomials coming from the quantum ﬁeld theory computations As we can see, all of these proofs are quite involved. On the other hand, our proofs of Miki’s and Faber- Pandharipande-Zagier’s identities follow from the polynomial identity (.), which in turn follows immediately the Fourier series expansion of γ m ( x ) in Theorems . and ., with ### Higher order Frobenius Euler and poly Bernoulli mixed type polynomials

Henceforth, F will denote both the algebra of formal power series in t and the vector space of all linear functionals on P, and so an element f (t) of F will be thought of as both a formal power series and a linear functional (see []). We shall call F the umbral algebra. The umbral calculus is the study of umbral algebra. The order o(f (t)) of a nonzero power series f (t) is the smallest integer k, for which the coeﬃcient of t k does not vanish. A series f (t) is called a delta series if o(f (t)) = , and an invertible series if o(f (t)) = . Let f (t), g(t) ∈ F . Then we have ### New results on higher order Daehee and Bernoulli numbers and polynomials

We derive a new matrix representation for higher-order Daehee numbers and polynomials, higher-order λ -Daehee numbers and polynomials, and twisted λ -Daehee numbers and polynomials of order k. This helps us to obtain simple and short proofs of many previous results on higher-order Daehee numbers and polynomials. Moreover, we obtain recurrence relations, explicit formulas, and some new results for these numbers and polynomials. Furthermore, we investigate the relation between these numbers and polynomials and Stirling, Nörlund, and Bernoulli numbers of higher-order. Some numerical results and program are introduced using Mathcad for generating higher-order Daehee numbers and polynomials. The results of this article generalize the results derived very recently by El-Desouky and Mustafa (Appl. Math. Sci. 9(73):3593-3610, 2015). ### On the Identities of Symmetry for the Generalized Bernoulli Polynomials Attached to of Higher Order

Simsek, “On p-adic twisted q-L-functions related to generalized twisted Bernoulli numbers,” Russian Journal of Mathematical Physics, vol.. Kim, “New approach to the complete sum of produ[r] ### Some identities of higher order Barnes type q Bernoulli polynomials and higher order Barnes type q Euler polynomials

Some identities of higher order Barnes type q Bernoulli polynomials and higher order Barnes type q Euler polynomials Jang et al Advances in Difference Equations (2015) 2015 162 DOI 10 1186/s13662 015[.] ### On the Generalized Genocchi Numbers and Polynomials of Higher Order

We first consider the q-extension of the generating function for the higher-order generalized Genocchi numbers and polynomials attached to χ. The purpose of this paper is to present a systemic study of some families of higher-order generalized q-Genocchi numbers and polynomials attached to χ by using the generating function of those numbers and polynomials. ### On the Identities of Symmetry for the Euler Polynomials of Higher Order

Then the values of E k n,ζ x at x 0 are called the ζ-Euler numbers of order k. When k 1, the polynomials or numbers are called the ζ-Euler polynomials or numbers. The purpose of this paper is to investigate some properties of symmetry for the multivariate p-adic fermionic integral on Z p . From the properties of symmetry for the multivariate p-adic fermionic integral ### q Bernoulli numbers and q Bernoulli polynomials revisited

This paper performs a further investigation on the q -Bernoulli numbers and q - Bernoulli polynomials given by Acikgöz et al. (Adv Differ Equ, Article ID 951764, 9, 2010), some incorrect properties are revised. It is point out that the generating function for the q -Bernoulli numbers and polynomials is unreasonable. By using the theorem of Kim (Kyushu J Math 48, 73-86, 1994) (see Equation 9), some new generating functions for the q -Bernoulli numbers and polynomials are shown. Mathematics Subject Classification (2000) 11B68, 11S40, 11S80