Journal of Applied Mathematics

Volume 2014 (2014), Article ID 806239, 7 pages

http://dx.doi.org/10.1155/2014/806239

Research Article

Zeros of Analytic Continued -Euler Polynomials and -Euler Zeta Function

C. S. Ryoo

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

Received 9 May 2014; Accepted 8 July 2014; Published 21 July 2014

Academic Editor: Carlos J. S. Alves

Copyright © 2014 C. S. Ryoo. 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 study that the -Euler numbers and -Euler polynomials are analytic continued to and . We investigate the new concept of dynamics of the zeros of analytic continued polynomials. Finally, we observe an interesting phenomenon of ‘‘scattering’’ of the zeros of .

1. Introduction

Recently, 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, many mathematicians can explore concepts much easier 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. Mathematicians have studied different kinds of the Euler, Bernoulli, Tangent, and Genocchi numbers and polynomials (see [1–14]). Numerical experiments of Bernoulli polynomials, Euler polynomials, Genocchi polynomials, and Tangent polynomials have been the subject of extensive study in recent years and much progress have 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. 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 real numbers, and denotes the set of complex numbers. In [11], we introduce the -Euler numbers and polynomials and investigate their properties. Let be a complex number with . By the meaning of (1) and (2), let us define the -Euler numbers and polynomials as follows (see [11]):

Observe that if , then and .

By using computer, the -Euler numbers can be determined explicitly. A few of them are

Theorem 1. For , one has

By Theorem 1, after some elementary calculations, we have

Since , by (5), we obtain

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

2. Analytic Continuation of Euler Numbers

In this section, we introduced the -Euler zeta function and Hurwitz -Euler zeta function. By -Euler zeta function, we consider the function as the analytic continuation of -Euler numbers.

From (1), we note that

By using the above equation, we are now ready to define -Euler zeta functions.

Definition 2. Let with Re. Consider

Observe that is a meromorphic function on . Clearly, (see [4, 7, 12]). Notice that the -Euler zeta function can be analytically continued to the whole complex plane, and these zeta function has the values of the -Euler numbers at negative integers.

Theorem 3. For , one has

Observe that function interpolates numbers at nonnegative integers.

By using (2), we note that

By (12), we are now ready to define the Hurwitz-type -Euler zeta functions.

Definition 4. Let with Re. Consider

Note that is a meromorphic function on . The relation between and is given by the following theorem.

Theorem 5. For , one has

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). Consider

In Figure 1, we choose . In fact, we can express in terms of , the derivative of . Consider

From the relation (17), we can define the other analytic continued half of -Euler numbers:

By (18), we obtain

The curve runs through the points and grows asymptotically as (see Figure 2).

Figure 1: The curve runs through the points of all except .

Figure 2: The curve runs through the points for .

3. Analytic Continuation of Euler Polynomials

In this section, we observe the analytic continued -Euler polynomials. Looking back at (9) and (18), for consistency with the definition of , -Euler polynomials should be analogously redefined as

Let be the gamma function. The analytic continuation can be then obtained as

where gives the integer part of , and so gives the fractional part.

By (21), we have analytic continuation of -Euler polynomials for . Consider

By using (22), we plot the deformation of the curve into the curve of via the real analytic continuation , , (see Figure 3).

Figure 3: The curve of , , and .

Next, we investigate the beautiful zeros of the by using a computer. We plot the zeros of for , , and (Figure 4).

(a)

(b)

(c)

(d)

Figure 4: Stacks of zeros of for .

In Figure 4, we observe that , , has reflection symmetry analytic complex functions (Figure 4). The obvious corollary is that the zeros of will also inherit these symmetries. Consider

where denotes complex conjugation.

Finally, we investigate the beautiful zeros of the by using a computer. We plot the zeros of for , , and (Figure 5). In Figure 5(a), we choose . In Figure 5(b), we choose . In Figure 5(c), we choose . In Figure 5(d), we choose .

(a)

(b)

(c)

(d)

Figure 5: Zeros of for .

Since

we obtain

The question is as follows: what happens with the reflexive symmetry (25), when one considers -Euler polynomials? Prove that , , has not reflection symmetry analytic complex functions (Figure 4). However, we observe that , , has reflection symmetry analytic complex functions (Figure 5).

Stacks of zeros of for , , forming a 3D structure are presented (Figure 6).

(a)

(b)

(c)

(d)

Figure 6: Stacks of zeros of for .

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).

Table 1: Numbers of real and complex zeros of .

Real zeros
Complex zeros

1.5
2
0

2.5
3
0

3.5
4
0

4.5
3
2

5.5
4
2

6.5
5
2

7.5
4
4

8.5
5
4

9.5
4
6

10
4
6

10.6
5
6

10.8
5
6

11
5
6

Next, we calculated an approximate solution satisfying , , . The results are given in Table 2.

Table 2: Approximate solutions of .

6
−0.471899, 0.51601, 1.52785, and 1.95754

6.5
−7.71893, −1.22075, −0.268389, 0.814309, and 0.947618

7
−0.842392, −0.0529263, and 0.947074

7.5
−8.73262, −1.55376, −0.838007, and 0.16246

8
−1.00171, −0.627539, 0.377875, and 1.37783

8.5
−9.74602, −1.62998, −1.46812, −0.406677, and 0.592636

9
−0.191227, 0.808773, and 1.80456

9.5
−10.7592, −0.975608, 0.0242117, and 1.0081

10
−0.758565, 0.239647, 1.23965, and 2.19578

10.6
−8.91801, −1.48957, −0.501829, 0.498175, and 1.39252

10.8
−5.27281, −1.41108, −0.415653, 0.58436, and 1.4511

11
−1.22287, −0.329477, 0.670523, 1.67056, and 2.5067

In Figure 7, we plot the real zeros of the -Euler polynomials for , , and (Figure 7).

Figure 7: Real zeros of .

-Euler polynomials are polynomials of degree . Thus, has zeros and has zeros. When discrete is analytic 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 Euler polynomials—the idea of looking at how the zeros move about in the complex plane as we vary the parameter .

To have a physical picture of the motion of the zeros in the complex plane, imagine that each time, as increases gradually and continuously by one, an additional real zero flies in from positive infinity along the real positive axis, gradually slowing down as if “it is flying through a viscous medium.”

For more studies and results on this subject you may see [5, 10–12].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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