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
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 .
Conjecture 7. Prove that the number of complex zeros of is where denotes taking the integer part.
Subsequently, much theoretical and computational work has been done, extending and testing these conjectures, with particular attention paid recently to certain refined conjectures for the analytic continued Euler polynomials (see [15]). By means of numerical experiments, we observe that , , has reflection symmetry in addition to the usual reflection symmetry analytic complex functions (Figures 1 and 2). The obvious corollary is that the zeros of will also inherit these symmetries: where denotes complex conjugation (see Figures 1 and 2). By Theorem 4 and Conjecture 5, the center of the structure of zeros is .
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The data concerning the numerical verification of Conjectures 6 and 7 are contained in Tables 1 and 2. See Table 1 for tabulated values of and . First, we investigate the beautiful zeros of the by using a computer. We plot the zeros of for and (Figure 1). In Figure 1(a), the zeros of structure are presented. In Figure 1(b), the real zeros of structure are presented for .
Stacks of zeros of for , forming a 3D structure, are presented (Figure 2). In Figure 2(a), we plot stacks of zeros of for . In Figure 2(b), we draw and axes but no axis in three dimensions. In Figure 2(c), we draw and axes but no axis in three dimensions. In Figure 2(d), we draw and axes but no axis in three dimensions.
Our numerical results for the numbers of real and complex zeros of are displayed (Table 1).
We observe a remarkably regular structure of the complex roots of Euler polynomials.
Next, we calculated an approximate solution satisfying , . The results are given in Table 2.
From (2), we have Comparing the coefficient of on both sides of (12), we get the following theorem.
Theorem 8. For any positive integer , one has
By (13), we have the following corollary.
Corollary 9. For , one has
3. Julia Set of the Euler Polynomials
In this section, we will present a series of diagrams showing the Julia set of the and its related Mandelbrot set. Computations of the Julia and Mandelbrot sets of the and observations of their properties are made. Let denote the set of complex numbers and . We generate graphic images using the software Mathematica. We define and construct orbits of points under the action of a complex function. Let be a complex function, with being a subset of . The iterates of are the functions , which are denoted by . If , then the orbit of under is the sequence .
Definition 10 (see [21]). The orbit of the point under the action of the function is said to be bounded if there exists such that for all . If the orbit is not bounded, it is said to be unbounded.
Plotting the orbit of length 3 of the point under the action of the function , we get the results shown in Figure 3.
When plotting the orbit of a point, it is a good idea to start by plotting a few points, in order to obtain an idea if the orbit is bounded or unbounded. We see that the orbit of is unbounded.
Definition 11 (see [21]). Let be a complex function, with being a subset of . The point is said to be a fixed point of if . One also says that is a fixed point of if as .
Definition 12 (see [21]). A fixed point of is an attractor if , a repeller if , and a neutral fixed point if .
As stated above, it can be proved that if is an attracting fixed point of , then there exists a neighbourhood of such that if the orbit of converges to . If is a repelling periodic point of , then there is a neighbourhood of such that if there are points in the orbit of which are not in . In the case of polynomials of degree greater than and some rational functions, is also called an attracting fixed point, as, for each such function, , there exists such that if then as .
Theorem 13 (fixed points of ). The Euler polynomials have one attractor fixed point (see Figure 4) atOur numerical results for fixed point of are displayed (Table 3). The results are obtained by Mathematica software.
Conjecture 14. For , the Euler polynomials have no attractor fixed point except for infinity.
Definition 15 (basins of attraction). Let be a complex function with attracting fixed point . The basin of attraction of under is defined to be the set
Definition 16 (basins of attraction of infinity). If infinity is an attracting fixed point of , then the basin of attraction of infinity is defined to be the set Let be a polynomial in of degree where .
Definition 17 (Julia set). The filled Julia set of is the set The Julia set of , , is the boundary of the filled Julia set of .
Consider the family of functions , where is a parameter. Julia sets can be divided into 2 classes. They are either connected or totally disconnected. Roughly speaking, a set which is connected is all in one piece (no breaks) while a set which is totally disconnected is like a cloud of dust particles, with its only connected components being points. If then both types of Julia sets occur and both types are generally aesthetically pleasing. If then the Julia set of is totally disconnected. This fact, together with the largeness of , produces Julia sets which are rather dispersed and spread out and so not very pleasing. We show (a) in Figure 5(a) the Julia set of which is connected; (b) in Figure 5(b) the Julia set of which is totally disconnected; (c) in Figure 5(c) the Julia set of which is totally disconnected.
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Examples of Julia sets for the family of are shown in Figure 6.
Newton tells us to consider the dynamical system is called the Newton iteration function of . It can be shown that the fixed points of are zeros of and that all fixed points of are attracting. may also have one or more attracting cycles. To obtain the Julia set of , plot, in white, the set of points whose orbits do not converge, and plot the set of remaining points whose orbits converge in a different coloring. The boundary of the set or of the set is the Julia set of .
Consider the Euler polynomial for . There are four distinct complex numbers, (), such that . Newton’s method provides a means to compute them. We obtain , , , and . Then Newton tells us to consider the dynamical system The general expectation is that a typical orbit , which starts from an initial “guess” , will converge to one of the roots of . If we choose close enough to then it is readily proved that
Given a point in the plane, we wish to find out if the orbit of under the action of does or does not converge to one of the roots of the equation, and if so, which one.
When is applied to , the orbit of under the action of is calculated until the absolute value of the last 2 iterations differs by an amount less than or until iterations have been carried out. The output is the last orbit value calculated. We construct a function which assigns one of 4 colors to each point in the plane, according to the outcome of . We assign the red, yellow, green, and blue to if its orbit converges to , respectively. We assign the black to it if the orbit does not converge to any of the points (Figure 7). The viewing windows are . In Figure 7, the red region represents part of the basin of attraction of .
As the Newton iteration function of is of degree 4, we plot, in white, the set of points whose orbits do not converge. The Julia set will be the boundary of the set of black points (see Figure 8). The viewing windows are .
A zoom-in on a version of Figure 8, in which the contours and contour lines are colored, is shown in Figure 9. The viewing windows are .
This color palette explains the coloring of Figure 9. For example, points which escape after 1 to 30 iterations are colored red to green. Points which do not escape after 30 iterations and so are taken to be in the filled Julia set are colored black (Figure 10).
The abovementioned rapid change can also be illustrated by applying the three-dimensional structure to the escape-time function (Figure 11). The viewing windows are . When is applied to , the orbit of under the action of is calculated until the absolute value of the last 2 iterations differs by an amount less than or until 30 iterations have been carried out. The output is the last orbit value calculated.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the NRF (National Research Foundation of Korea) Grant funded by the Korean Government (NRF-2013-Fostering Core Leaders of the Future Basic Science Program).