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]

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[.]

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[.]

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

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:

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.

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**.

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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

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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**.

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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]

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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**.

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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.

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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

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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

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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).

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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 Jang et al Advances in Difference Equations (2015) 2015 162 DOI 10 1186/s13662 015[.]

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**.

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

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