Abstract and Applied Analysis

Volume 2013 (2013), Article ID 459763, 6 pages

http://dx.doi.org/10.1155/2013/459763

Research Article

Some Identities on the High-Order -Euler Numbers and Polynomials with Weight 0

Jongsung Choi,¹ Hyun-Mee Kim,² and Young-Hee Kim¹

¹Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

²Department of General Education-Mathematics, Kookmin University, Seoul 136-702, Republic of Korea

Received 7 February 2013; Accepted 2 April 2013

Academic Editor: Chun-Gang Zhu

Copyright © 2013 Jongsung Choi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We construct the th order nonlinear ordinary differential equation related to the generating function of -Euler numbers with weight 0. From this, we derive some identities on -Euler numbers and polynomials of higher order with weight 0.

1. Introduction

As a well-known definition, the Euler polynomial is given by

In the special case, , is the th Euler number.

From (1), we note that

with the usual convention of replacing by (see [1–16]).

In the viewpoint of the -extension of (1) and (2), let us consider the following -Euler number and polynomial:

with the usual convention of replacing by .

Equation (3) is called the generating function of -Euler polynomial with weight . In the case , is the th -Euler number with weight (see [5, 11]).

Throughout this paper, let be a complex number with . As , we obtain (1) and (2) from (3) and (4).

The generating function of Eulerian polynomial is defined by

where with . In the special case, , is called the th Eulerian number (see [1–3]). Sometimes that is called the th Frobenius-Euler number (see [9–11, 15]).

From (1) and (5), we note that . From (5), we have

where is Kronecker symbol (see [9–11]).

For , the -Euler polynomial of order is defined by the generating function as follows:

In the special case, , is called the th -Euler number of order with weight (see [5, 11]).

In [9], Kim derived some identities between the sums of products of Frobenius-Euler polynomials and Frobenius-Euler polynomials of higher order. The main idea is to construct nonlinear ordinary differential equations with respect to which are closely related to the generating function of Frobenius-Euler polynomial. In [3], Choi considered nonlinear ordinary differential equations with respect to not .

In this paper, we construct nonlinear ordinary differential equations with respect to . The purpose of this paper is to give some new identities on the high order -Euler numbers and polynomials with weight by using the differential equations of .

2. Construction of Nonlinear Differential Equations

We define

From (7) and (8), we note that

By differentiating (8) with respect to , we get

By differentiating (10) with respect to , we get

where .

By the derivative of (11) with respect to , we have

Continuing this process, we get

Let us consider the derivative of (13) with respect to to find the coefficient in (13).

By (10), we get

From (13) and (14), we get

where .

By comparing coefficients on both sides of (15), we obtain the following recurrence relations:

for and .

From the first part of (16), we have

By (10) and (13), we have

From (18) and (19), we get

From the second part of (16), we have

To find in (13) from (17), we set

where (see [9]).

From (17) and (22), we have

From the left hand side of (23), we have

where . From the first term of the right hand side of (23), we have

From the second term of the right hand side of (23), we have

where .

From (22)–(26), we obtain the following initial value problem quasilinear first-order partial differential equation:

We consider Cauchy problem for the following first-order quasilinear partial differential equation:

where is some interval.

We know that (28) has a unique solution under some conditions as follows.

Theorem A (see [17, page 65]). Suppose that , and are of class in a domain of containing the point and suppose that

Then in a neighborhood of there exists a unique solution of (28) at every point of initial curve contained in .

Since (27) satisfies (29) and regularity conditions, there exists a unique solution of (27).

It is customary to write (27) in the form

Since is separable, we get

is a solution of partial differential equation of (27).

From (30), we get the linear equation

By the integrating factor method, we have

The exponential integral is defined by

where is Euler constant.

is another solution of partial differential equation of (27), and and are linearly independent.

From the parameterized initial conditions (31), (33), and (34), we get

Thus, from (35) and (36), we obtain the following unique solution of (27):

Moreover, if we choose another initial condition

from (20) and (22), then (37) satisfies it.

We note that

By (37) and (39), we get

It is known that

In the case of in (40), from (41), we get

By (40) and (41), we get

where . Thus, by (22) and (43), we get

Therefore, by (13) and (44), we obtain the following theorem.

Theorem 1. For with and , one can consider the following nonlinear th order ordinary differential equation with respect to :

where and . Then is a solution of (45).

Let us define . Then we obtain the following corollary.

Corollary 2. For , one considers

Then is a solution of (46).

3. Identities on the High-Order -Euler Numbers and Polynomials with Weight 0

From (3), (7), and (8), we get

From (47), we note that

Therefore, by (47), (48), and (45), we obtain the following theorem.

Theorem 3. For and , one has

From (48), we get

Therefore, by (7), (47), and (50), we obtain the following corollary.

Corollary 4. For and , one has

From (3) and (7), we get

Therefore, by (49) and (52), we obtain the following corollary.

Corollary 5. For and , one has

Acknowledgment

The present research has been conducted by the research grant of Kwangwoon University in 2013.

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