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Abstract and Applied Analysis

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

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

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

^{1}Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea^{2}Department 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.

#### References

- L. Carlitz, “Eulerian numbers and polynomials,”
*Mathematics Magazine*, vol. 32, pp. 247–260, 1959. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Carlitz, “The product of two Eulerian polynomials,”
*Mathematics Magazine*, vol. 36, no. 1, pp. 37–41, 1963. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Choi, “A note on Eulerian polynomials of higher order,”
*Journal of the Chungcheong Mathematical Society*, vol. 26, no. 1, pp. 191–196, 2013. - J. Choi, T. Kim, and Y. H. Kim, “A recurrence formula for
*q*-Euler numbers of higher order,”*Proceedings of the Jangjeon Mathematical Society*, vol. 13, no. 3, pp. 321–326, 2010. View at Zentralblatt MATH · View at Scopus - J. Choi, T. Kim, and Y.-H. Kim, “A note on the $q$-analogues of Euler numbers and polynomials,”
*Honam Mathematical Journal*, vol. 33, no. 4, pp. 529–534, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - D. S. Kim, “Identities of symmetry for generalized Euler polynomials,”
*International Journal of Combinatorics*, vol. 2011, Article ID 432738, 12 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - D. S. Kim, T. Kim, J. Choi, and Y. H. Kim, “Identities involving $q$-Bernoulli and $q$-Euler numbers,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 674210, 10 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H.-M. Kim, J. Choi, and T. Kim, “On the extended $q$-Euler numbers and polynomials of higher-order with weight,”
*Honam Mathematical Journal*, vol. 34, no. 1, pp. 1–9, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - T. Kim, “Identities involving Frobenius-Euler polynomials arising from non-linear differential equations,”
*Journal of Number Theory*, vol. 132, no. 12, pp. 2854–2865, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - T. Kim and J. Choi, “A note on the product of Frobenius-Euler polynomials arising from the $p$-adic integral on ${\mathbb{Z}}_{p}$,”
*Advanced Studies in Contemporary Mathematics*, vol. 22, no. 2, pp. 215–223, 2012. View at MathSciNet - T. Kim and J. Choi, “On the $q$-Euler numbers and polynomials with weight 0,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 795304, 7 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - H. Ozden, I. N. Cangul, and Y. Simsek, “Multivariate interpolation functions of higher-order $q$-Euler numbers and their applications,”
*Abstract and Applied Analysis*, vol. 2008, Article ID 390857, 16 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - H. Ozden and Y. Simsek, “A new extension of $q$-Euler numbers and polynomials related to their interpolation functions,”
*Applied Mathematics Letters*, vol. 21, no. 9, pp. 934–939, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Simsek, “Complete sum of products of $(h,q)$-extension of Euler polynomials and numbers,”
*Journal of Difference Equations and Applications*, vol. 16, no. 11, pp. 1331–1348, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Simsek, “Generating functions for
*q*-Apostol type Frobenius-Euler numbers and polynomials,”*Axioms*, vol. 1, no. 3, pp. 395–403, 2012. View at Publisher · View at Google Scholar - Y. Simsek, “Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications,”
*Fixed Point Theory and Applications*, vol. 2013, article 87, 2013. View at Publisher · View at Google Scholar - C. Zachmanoglou and D. W. Thoe,
*Introduction to Partial Differntial Equations with Applications*, The Williams and Wilkins company, Baltimore, Md, USA, 1976.