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 Bernoullipolynomials 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 higherorder 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 Bernoullipolynomials. 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.
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
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 higherorder.
We define the twisted q-Bernoullipolynomials and the twisted generalized q-Bernoulli polynomi- als attached to χ of higherorder 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.
This paper gives a new generalization of higherorder Daehee and Bernoulli numbers and polynomials. We define the multiparameter higherorder 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 higherorder Dae- hee and Bernoulli numbers and polynomials are deduced.
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
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
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).
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 - Bernoullipolynomials 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