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

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, andmake 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, andGenocchi numbers and polynomials. Recently, many authors have studied in the area of the q-analogues of these numbers and polynomials (see [1–20]). Using computer, a realistic study for the zeros of Euler polynomials E n (x) is very interesting.Themain purpose of this paper is to observe an interesting phenomenon of “scattering” of the zeros of the Euler polynomials E n (x) in complex plane. Throughout this paper, we always make use of the following notations: N = {1, 2, 3, . . .} denotes the set of natural numbers, N0 = {0, 1, 2, 3, . . .} denotes the set of nonnegative integers,Z denotes the set of integers,R denotes the set of real numbers, and C denotes the set of complex numbers. The classical Euler numbers E n and Euler polynomials


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][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][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: N = {1, 2, 3, . ..} denotes the set of natural numbers, N 0 = {0, 1, 2, 3, . ..} denotes the set of nonnegative integers, Z denotes the set of integers, R denotes the set of real numbers, and C 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.
Theorem 1.For  ∈ N 0 , one has By Theorem 1, after some elementary calculations, we get By Theorem 1, we have Since   (0) =   , by (5), we have the following theorem.
Theorem 2. For  ∈ N, one has Then, it is easy to deduce that   () are polynomials of degree .Here is the list of the first Euler polynomials:

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.
In [13,15], we made a series of the following conjectures.
Since  is the degree of the polynomial   (), the number of real zeros    () lying on the real plane Im() = 0 is then    () =  −    () , where    () denotes complex zeros.See Table 1 for tabulated values of    () and    () .

Conjecture 7. Prove that the number of complex zeros 𝐶
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   (),  ∈ C, has Re() = 1/2 reflection symmetry in addition to the usual Im() = 0 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 1/2.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  = 70 and  ∈ C (Figure 1).In Figure 1(a), the zeros of  70 () structure are presented.In Figure 1(b), the real zeros of   () structure are presented for 1 ≤  ≤ 70.
Stacks of zeros of   () for 1 ≤  ≤ 70, forming a 3D structure, are presented (Figure 2).In Figure 2(a), we plot stacks of zeros of   () for 1 ≤  ≤ 70.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   (),  ∈ R. 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  ∈ N, one has

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 C denote the set of complex numbers and C ∞ = C ∪ {∞}.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 C. The iterates of  are the functions ,  ∘ ,  ∘  ∘ , . .., which are denoted by  1 ,  2 ,  3 , . ... If  ∈ C, 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  ∈ R such that |  ()| ≤  for all  ∈ N. If the orbit is not bounded, it is said to be unbounded.
Plotting the orbit of length 3 of the point 1 + 0.5 under the action of the function  3 (), 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  3 () is unbounded.
Definition 11 (see [21]).Let  :  →  be a complex function, with  being a subset of C. The point  is said to be a fixed point of  if () = .One also says that ∞ is a fixed point of 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 0 and some rational functions, ∞ is also called an attracting fixed point, as, for each such function, , there exists  > 0 such that if || >  then   () → ∞ as  → ∞.
Theorem 13 (fixed points of   ()).The Euler polynomials  3 () + 0.2 + 0.5 have one attractor fixed point (see Figure 4) at Our numerical results for fixed point of   () are displayed (Table 3).The results are obtained by Mathematica software.Conjecture 14.For  > 3, 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  : C → C be a polynomial in  of degree > 2 where  ∈ C. 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 { 3 () :  →  3 () +  |  ∈ C}, 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 Examples of Julia sets for the family of { 5 () :  →  5 () +  |  ∈ C}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  4 () for  ∈ C.There are four distinct complex numbers,   ( = 1, 2, 3, 4), such that  4 (  ) = 0. Newton's method provides a means to compute them.We obtain  1 = 0,  2 = 2,  3 = (1/2)(1 − √ 5), and  4 = (1/2)(1 + √ 5).Then Newton tells us to consider the dynamical system The general expectation is that a typical orbit {  ()}, which starts from an initial "guess"  0 ∈ C, will converge to one of the roots of  4 ().If we choose  0 close enough to   then it is readily proved that lim Given a point  0 in the plane, we wish to find out if the orbit of  0 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  0 , the orbit of  0 under the action of  is calculated until the absolute value of the last 2 iterations differs by an amount less than 10 −10 or until 30 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  0 if its orbit converges to 0, 1, (1/2)(1 − √ 5), (1/2)(1 +  √ 5), respectively.We assign the black to it if the orbit does not converge to any of the points (Figure 7).The viewing windows are {(, ) : −6 ≤  ≤ 6, −6 ≤  ≤ 6}.In Figure 7, the red region represents part of the basin of attraction of  1 = 0.As the Newton iteration function of  4 () 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 {(, ) : −5.5 ≤  ≤ 5.5, −5.5 ≤  ≤ 5.5}.
A zoom-in on a version of Figure 8, in which the contours and contour lines are colored, is shown in Figure 9.The 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 escapetime function (Figure 11).The viewing windows are {(, ) : −5.5 ≤  ≤ 5.5, −5.5 ≤  ≤ 5.5}.When () is applied to  0 , the orbit of  0 under the action of  is calculated until the     absolute value of the last 2 iterations differs by an amount less than 10 −10 or until 30 iterations have been carried out.The output is the last orbit value calculated.

Figure 7 :
Figure 7: Orbit of  0 under the action of  for  4 ().

Table 1 :
Numbers of real and complex zeros of   ().

Table 3 :
Numbers of attractor, repeller, and neutral fixed points of   ().