Research Article | Open Access
C. S. Ryoo, "Zeros of Analytic Continued -Euler Polynomials and -Euler Zeta Function", Journal of Applied Mathematics, vol. 2014, Article ID 806239, 7 pages, 2014. https://doi.org/10.1155/2014/806239
Zeros of Analytic Continued -Euler Polynomials and -Euler Zeta Function
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 .
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 , 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 ): 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.
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).
3. Analytic Continuation of Euler Polynomials
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
Next, we investigate the beautiful zeros of the by using a computer. We plot the zeros of for , , and (Figure 4).
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 .
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).
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 , , . The results are given in Table 2.
-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.”
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
- R. Ayoub, “Euler and the zeta function,” The American Mathematical Monthly, vol. 81, pp. 1067–1086, 1974.
- A. Bayad, “Modular properties of elliptic Bernoulli and Euler functions,” Advanced Studies in Contemporary Mathematics, vol. 20, pp. 389–401, 2010.
- J. Y. Kang, H. Y. Lee, and N. S. Jung, “Some relations of the twisted q-Genocchi numbers and polynomials with weight α and weak weight β,” Abstract and Applied Analysis, vol. 2012, Article ID 860921, 9 pages, 2012.
- M. Kim and S. Hu, “On -adic Hurwitz-type Euler zeta functions,” Journal of Number Theory, vol. 132, no. 12, pp. 2977–3015, 2012.
- T. Kim, C. S. Ryoo, L. C. Jang, and S. H. Rim, “Exploring the q-Riemann zeta function and q-Bernoulli polynomials,” Discrete Dynamics in Nature and Society, no. 2, pp. 171–181, 2005.
- T. Kim and S. H. Rim, “Generalized Carlitz's Euler Numbers in the p-adic number field,” Advanced Studies in Contemporary Mathematics, vol. 2, pp. 9–19, 2000.
- T. Kim, “Euler numbers and polynomials associated with zeta functions,” Abstract and Applied Analysis, vol. 2008, Article ID 581582, 11 pages, 2008.
- H. Ozden and Y. Simsek, “A new extension of -Euler numbers and polynomials related to their interpolation functions,” Applied Mathematics Letters, vol. 21, no. 9, pp. 934–939, 2008.
- S. Rim, K. H. Park, and E. J. Moon, “On Genocchi numbers and polynomials,” Abstract and Applied Analysis, vol. 2008, Article ID 898471, 7 pages, 2008.
- C. S. Ryoo, T. Kim, and R. P. Agarwal, “A numerical investigation of the roots of -polynomials,” International Journal of Computer Mathematics, vol. 83, no. 2, pp. 223–234, 2006.
- C. S. Ryoo, “A numerical computation of the roots of -Euler polynomials,” Journal of Computational Analysis and Applications, vol. 12, no. 1-A, pp. 148–156, 2010.
- C. S. Ryoo, “Analytic continuation of Euler polynomials and the Euler zeta function,” Discrete Dynamics in Nature and Society, vol. 2014, Article ID 568129, 6 pages, 2014.
- Y. Simsek, “Generating functions of the twisted Bernoulli numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 251–257, 2008.
- Y. Simsek, “Twisted (h,q)-Bernoulli numbers and polynomials related to twisted (h,q)-zeta function and L-function,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 790–804, 2006.
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