Discrete Dynamics in Nature and Society

Volume 2015 (2015), Article ID 174173, 9 pages

http://dx.doi.org/10.1155/2015/174173

## A Numerical Investigation on the Structure of the Zeros of Euler Polynomials

Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea

Received 13 January 2015; Accepted 17 May 2015

Academic Editor: Guang Zhang

Copyright © 2015 C. S. Ryoo and J. Y. Kang. 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

Using numerical investigation, we observe the behavior of complex roots of the Euler polynomials . By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the Euler polynomials . Finally, we show the Julia set of Newton iteration function .

#### 1. Introduction

The computing environment would make more and more rapid progress and there has been increasing interest in solving mathematical problems with the aid of computers. By using software, mathematicians can explore concepts much more easily than in the past. The ability to create and manipulate figures on the computer screen enables mathematicians to quickly visualize and produce many problems, examine properties of the figures, look for patterns, and make conjectures. This capability is especially exciting because these steps are essential for most mathematicians to truly understand even basic concept. Numerical experiments of Bernoulli polynomials, Euler polynomials, and Genocchi polynomials have been the subject of extensive study in recent year and much progress has been made both mathematically and computationally. Mathematicians have studied different kinds of the Euler, Bernoulli, Tangent, and Genocchi numbers and polynomials. Recently, many authors have studied in the area of the -analogues of these numbers and polynomials (see [1–20]). Using computer, a realistic study for the zeros of Euler polynomials is very interesting. The main purpose of this paper is to observe an interesting phenomenon of “scattering” of the zeros of the Euler polynomials in complex plane. Throughout this paper, we always make use of the following notations: denotes the set of natural numbers, denotes the set of nonnegative integers, denotes the set of integers, denotes the set of real numbers, and denotes the set of complex numbers.

The classical Euler numbers and Euler polynomials are usually defined by the following generating functions: where we use the technique method notation by replacing by symbolically.

Clearly, . These polynomials and numbers play important roles in many different areas of mathematics such as combinatorics, number theory, special function, and analysis, and numerous interesting results for them have been explored (see [2, 3, 7–9, 13, 15, 18–20]). The following elementary properties of Euler polynomials are readily derived from (1) and (2). We, therefore, choose to omit the details involved. For more studies and results in this subject we may see [2, 3, 8, 13, 15, 17–20].

Theorem 1. *For , one has *

*By Theorem 1, after some elementary calculations, we get By Theorem 1, we have Since , by (5), we have the following theorem.*

*Theorem 2. For , one has *

*Then, it is easy to deduce that are polynomials of degree . Here is the list of the first Euler polynomials: *

*2. The Phenomenon of Scattering of Zeros*

*2. The Phenomenon of Scattering of Zeros*

*In this section, an interesting phenomenon of scattering of zeros of is observed. By (2), we obtain *

*Hence we have the following theorem.*

*Theorem 3 (see [13, 15]). For , one has *

*By (9), we also have the following theorem.*

*Theorem 4. For , if , then ; if , then .*

*In [13, 15], we made a series of the following conjectures.*

*Conjecture 5. Prove that , , has reflection symmetry in addition to the usual reflection symmetry analytic complex functions.*

*Conjecture 6. Prove that has distinct solutions.*

*We find a counterexample of Conjecture 6. Let . Then there are five numbers, (), such that . That is, we obtain , , , , and . Hence, Conjecture 6 is not true for all . Using computers, many more values of have been checked. It still remains unknown if the conjecture fails or holds for any value .*

*Since is the degree of the polynomial , the number of real zeros lying on the real plane is then , where denotes complex zeros. See Table 1 for tabulated values of and .*