Analytic Continuation of Euler Polynomials and the Euler Zeta Function

Throughout this paper, Z, R, and C will denote the ring of integers, the field of real numbers, and the complex numbers, respectively. Recently, many mathematicians have studied different kinds of the Euler, Bernoulli, andGenocchi numbers and polynomials (see [1–19]). 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 formostmathematicians 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 years and much progress has been made both mathematically and computationally. Using computer, a realistic study for Euler polynomialsE n (x) is very interesting. It is the aim of this paper to observe an interesting phenomenon of “scattering” of the zeros of the Euler polynomials E n (x) in complex plane. First, we introduce the Euler numbers and Euler polynomials. As a well-known definition, the Euler numbers E n are defined by


Introduction
Throughout this paper, Z, R, and C will denote the ring of integers, the field of real numbers, and the complex numbers, respectively.Recently, many mathematicians have studied different kinds of the Euler, Bernoulli, and Genocchi numbers and polynomials (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]).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 years and much progress has been made both mathematically and computationally.Using computer, a realistic study for Euler polynomials   () is very interesting.It is the aim of this paper to observe an interesting phenomenon of "scattering" of the zeros of the Euler polynomials   () in complex plane.First, we introduce the Euler numbers and Euler polynomials.As a well-known definition, the Euler numbers   are defined by Here is the list of the first Euler numbers: ( The Euler polynomials   () are defined by the generating function: where we use the technique method notation by replacing ()  by   () symbolically.Because an important relation follows: Then, it is easy to deduce that   () are polynomials of degree .Here is the list of the first Euler polynomials:

Generating Euler Polynomials and Numbers
we have the following theorem.
Notice that the Euler zeta function can be analytically continued to the whole complex plane, and these zeta functions have the values of the Euler numbers at negative integers.That is, Euler numbers are related to the Euler zeta function as Definition 3. We define the Hurwitz zeta function   (, ) for  ∈ C with Re() > 0 and  ∈ R with 0 ≤  < 1 by see [3,[7][8][9][10]12].
Euler polynomials are related to the Hurwitz zeta function as We now consider the function () as the analytic continuation of Euler numbers.From the above analytic continuation of Euler numbers, we consider All the Euler numbers   agree with (), the analytic continuation of Euler numbers evaluated at  (see Figure 1), except In fact, we can express   () in terms of    (), the derivative of   (), From relation (16), we can define the other analytic continued half of Euler numbers as By (17), we have lim The curve () runs through the points  − = (−) and grows ∼ −2 asymptotically as − → ∞ (see Figure 2).

Analytic Continuation of Euler Polynomials
Looking back at (1) and (3), we can see that the sign convention of  0 was actually arbitrary.Equation (15) suggests that consistent definition of Euler numbers should really have been which only changes the sign in the conventional definition of the only nonzero even Euler numbers,  0 , from  0 = 1 to  0 = (0) = −1.By using Cauchy product, we have For consistency with the redefinition of   = () in ( 19), Euler polynomials should be analogously redefined as The analytic continuation can be then obtained as where [] gives the integer part of , and so −[] gives the fractional part.By (22), we obtain analytic continuation of Euler polynomials: By using (23), we plot the deformation of the curve (2, ) into the curve of (3, ) via the real analytic continuation (, ), 2 ≤  ≤ 3,  ∈ R (see Figure 3).
Next, we investigate the beautiful zeros of the (, ) by using a computer.We plot the zeros of (, ) for  = 9, 9.6, 9.8, 10 and  ∈ C (Figure 4).
Our numerical results for approximate solutions of real zeros of (, ) are displayed.We observe a remarkably regular structure of the complex roots of Euler polynomials.We hope to verify a remarkably regular structure of the complex roots of Euler polynomials (Table 1).Next, we calculated an approximate solution satisfying (, ),  ∈ R. The results are given in Table 2.   Euler polynomials   () are polynomials of degree .Thus,   () has  zeros and  +1 () has  + 1 zeros.When discrete  is analytically continued to continuous parameter , it naturally leads to the following question: how does (, ), the analytic continuation of   (), pick up an additional zero as  increases continuously by one?
This introduces the exciting concept of the dynamics of the zeros of analytic continued polynomials, the idea of looking at how the zeros move about in the  complex plane as we vary the parameter .For more studies and results in this subject you may see [11,[14][15][16].

Figure 1 : 4 Figure 2 :
Figure 1: The curve () runs through the points of all   except  0 .

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