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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 459763, 6 pages
Some Identities on the High-Order -Euler Numbers and Polynomials with Weight 0
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.
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.
As a well-known definition, the Euler polynomial is given by In the special case, , is the th Euler number.
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]).
In , 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 , 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
By the derivative of (11) with respect to , we have Continuing this process, we get
By comparing coefficients on both sides of (15), we obtain the following recurrence relations: for and .
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 .
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 .
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
Theorem 3. For and , one has
Corollary 4. For and , one has
Corollary 5. For and , one has
The present research has been conducted by the research grant of Kwangwoon University in 2013.
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