Taekyun Kim

Some identities on the q-Euler polynomials of higher order and q-stirling numbers by the fermionic p-adic integral on ℤp

A systemic study of some families of q-Euler numbers and families of polynomials of Nörlund type is presented by using the multivariate fermionic p-adic integral on ℤp. The study of these higher-order q-Euler numbers and polynomials yields an interesting q-analog of identities for Stirling numbers.

q-Euler numbers and polynomials associated with p-adic q-integrals

Abstract The main purpose of this paper is to present a systemic study of some families of multiple q-Euler numbers and polynomials. In particular, by using the q-Volkenborn integration on ℤp, we construct p-adic q-Euler numbers and polynomials of higher order. We also define new generating functions of multiple q-Euler numbers and polynomials. Furthermore, we construct Euler q-Zeta function.

On the multiple q-Genocchi and Euler numbers

The purpose of this paper is to present a systematic study of some families of multiple q-Genocchi and Euler numbers by using the multivariate q-Volkenborn integral (= p-adic q-integral) on ℤp. The investigation of these q-Genocchi numbers and polynomials of higher order leads to interesting identities related to these objects. The results of the present paper cover earlier results concerning ordinary q-Genocchi numbers and polynomials.

Symmetry p-adic invariant integral on ℤ p for Bernoulli and Euler polynomials

The main purpose of this paper is to investigate several further interesting properties of symmetry for the p-adic invariant integrals on ℤ p . From these symmetry, we can derive many interesting recurrence identities for Bernoulli and Euler polynomials. Finally we introduce the new concept of symmetry of fermionic p-adic invariant integral on ℤ p . By using this symmetry of fermionic p-adic invariant integral on ℤ p , we will give some relations of symmetry between the power sum polynomials and Euler numbers. The relation between the q-Bernoulli polynomials and q-Dedekind type sums which discussed in Y. Simsek (q-Dedekind type sums related to q-zeta function and basic L-series, J. Math. Anal. Appl. 318 (2006), pp. 333–351) can be also derived by using the properties of symmetry of fermionic p-adic integral on ℤ p .

Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on ℤp

The objective of the paper is to indicate a symmetry of the multivariate p-adic invariant integral on ℤp, which leads to a relation between the power sum polynomials and higher-order Euler polynomials.

Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials

In this paper, we give relations involving values of q-Bernoulli, q-Euler, and Bernstein polynomials. Using these relations, we obtain some interesting identities on the q-Bernoulli, q-Euler, and Bernstein polynomials.

New approach to q-Euler polynomials of higher order

In this paper, we present new generating functions related to q-Euler numbers and polynomials of higher order. Using those generating functions, we present new identities involving q-Euler numbers and polynomials of higher order.In this paper we give new identities involving q-Euler polynomials of higher order.

Barnes-type multiple q-zeta functions and q-Euler polynomials

The purpose of this paper is to present a systemic study of some families of multiple q-Euler numbers and polynomials and we construct multiple q-zeta functions which interpolate multiple q-Euler numbers at a negative integer. This is a partial answer to the open question in a previous publication (see Kim et al 2001 J. Phys. A: Math. Gen. 34 7633–8).

Euler Numbers and Polynomials Associated with Zeta Functions

For , the Euler zeta function and the Hurwitz-type Euler zeta function are defined by , and . Thus, we note that the Euler zeta functions are entire functions in whole complex -plane, and these zeta functions have the values of the Euler numbers or the Euler polynomials at negative integers. That is, , and . We give some interesting identities between the Euler numbers and the zeta functions. Finally, we will give the new values of the Euler zeta function at positive even integers.

q-Generalized Euler numbers and polynomials

Recently, B. A. Kupershmidt constructed reflection symmetries of q-Bernoulli polynomials (see [12]). In this paper, we study new q-extensions of Euler numbers and polynomials by using the method of Kupershmidt. We also investigate the properties of symmetries of these q-Euler polynomials by using q-derivatives and q-integrals.