Lee-Chae Jang

Some Relationships between the Analogs of Euler Numbers and Polynomials

We construct new twisted Euler polynomials and numbers. We also study the generating functions of the twisted Euler numbers and polynomials associated with their interpolation functions. Next we construct twisted Euler zeta function, twisted Hurwitz zeta function, twisted Dirichlet-Euler numbers and twisted Euler polynomials at non-positive integers, respectively. Furthermore, we find distribution relations of generalized twisted Euler numbers and polynomials. By numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the twisted-Euler polynomials. Finally, we give a table for the solutions of the twisted-Euler polynomials.

A note on Euler number and polynomials

We investigate some properties of non-Archimedean integration which is defined by Kim. By using our results in this paper, we can give an answer to the problem which is introduced by I.-C. Huang and S.-Y. Huang in 1999.

Note on q -Extensions of Euler Numbers and Polynomials of Higher Order

In 2007, Ozden et al. constructed generating functions of higher-order twisted (h, q)-extension of Euler polynomials and numbers, by using p-adic, q-deformed fermionic integral on . By applying their generating functions, they derived the complete sums of products of the twisted (h, q)-extension of Euler polynomials and numbers. In this paper, we consider the new q-extension of Euler numbers and polynomials to be different which is treated by Ozden et al. From our q-Euler numbers and polynomials, we derive some interesting identities and we construct q-Euler zeta functions which interpolate the new q-Euler numbers and polynomials at a negative integer. Furthermore, we study Barnes-type q-Euler zeta functions. Finally, we will derive the new formula for sums of products of q-Euler numbers and polynomials by using fermionic q-adic, q-integral on .

A Note on Nörlund-Type Twisted -Euler Polynomials and Numbers of Higher Order Associated with Fermionic Invariant -Integrals

We define multiple Nörlund-type twisted -Euler polynomials and numbers and give interpolation functions of multiple Nörlund-type twisted -Euler polynomials at negative integers. Furthermore, we investigate some identities related to these polynomials and interpolation functions.

On the distribution of the -Euler polynomials and the -Genocchi polynomials of higher order.

In 2007 and 2008, Kim constructed the -extension of Euler and Genocchi polynomials of higher order and Choi-Anderson-Srivastava have studied the -extension of Euler and Genocchi numbers of higher order, which is defined by Kim. The purpose of this paper is to give the distribution of extended higher-order -Euler and -Genocchi polynomials.

Kummer congruence for the Bernoulli numbers of higher order

The authors studied the properties of Bernoulli numbers of higher order [Appl. Math. Comput., in press; Bull. Aust. Math. 65 (2002) 59]. For q=1, we can also find their results [Proc. Jangjeon Math. Soc. 1 (2000) 97; Arch. Math. 76 (2001) 190; Proc. Jangjeon Math. Soc. 1 (2000) 161; Adv. Stud. Contemp. Math. 2 (2000) 9; Proc. Jangjeon Math. Soc. 2 (2001) 23; J. Math. Phys. A 34 (2001) L643; Proc. Jangjeon Math. Soc. 2 (2001) 19; Proc. Jangjeon Math. Soc. 2 (2001) 9; Proc. Jangjeon Math. Soc. 3 (2001) 63]. The authors suggested the question to inquire the proof of Kummer congruence for Bernoulli numbers of higher order [Appl. Math. Comput., in press]. In this paper we give a proof of Kummer type congruence for the Bernoulli numbers of higher order, which is an answer to a part of the question in [Appl. Math. Comput., in press].

A note on generalized Euler numbers and polynomials

In this paper we construct new generalized Euler polynomials and generalized Euler numbers attached to χ. We investigate some of the properties that are related to generalized Euler polynomials. We also derive the existence of a specific interpolation function that interpolates generalized Euler polynomials at negative integers. Finally, we investigate computationally the roots of the generalized Euler polynomials E n, χ(x) for values of the index n.

Note on the -Extension of Barnes' Type Multiple Euler Polynomials

We construct the -Euler numbers and polynomials of higher order, which are related to Barnes type multiple Euler polynomials. We also derive many properties and formulae for our -Euler polynomials of higher order by using the multiple integral equations on .

A Note on Hölder Type Inequality for the Fermionic -Adic Invariant -Integral

The purpose of this paper is to find Hölder type inequality for the fermionic -adic invariant -integral which was defined by Kim (2008).

A Note on Symmetric Properties of the Twisted -Bernoulli Polynomials and the Twisted Generalized -Bernoulli Polynomials

We define the twisted -Bernoulli polynomials and the twisted generalized -Bernoulli polynomials attached to of higher order and investigate some symmetric properties of them. Furthermore, using these symmetric properties of them, we can obtain some relationships between twisted -Bernoulli numbers and polynomials and between twisted generalized -Bernoulli numbers and polynomials.