Analytic Continuation of Euler Polynomials and the Euler Zeta Function
We study that the Euler numbers and Euler polynomials are analytically continued to and . We investigate the new concept of dynamics of the zeros of analytica continued polynomials. Finally, we observe an interesting phenomenon of “scattering” of the zeros of .
Throughout this paper, , , and 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 [1–19]). 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:
2. Generating Euler Polynomials and Numbers
Since we have the following theorem.
Theorem 1. For , one has
Definition 2. For with , define the Euler zeta function by see [7–10].
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 with and with by see [3, 7–10, 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),
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
The curve runs through the points and grows ~−2 asymptotically as (see Figure 2).
3. Analytic Continuation of Euler Polynomials
Looking back at (1) and (3), we can see that the sign convention of 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, , from to .
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 into the curve of via the real analytic continuation , , (see Figure 3).
Next, we investigate the beautiful zeros of the by using a computer. We plot the zeros of for and (Figure 4).
In Figure 4(a), we choose . In Figure 4(b), we choose . In Figure 4(c), we choose . In Figure 4(d), we choose .
Since we obtain
Hence, we have the following theorem.
Theorem 4. If , then , for .
The question is, what happens with the reflexive symmetry (25) when one considers Euler polynomials? Prove that , , has reflection symmetry in addition to the usual reflection symmetry analytic complex functions. However, we observe that , , does not have reflection symmetry analytic complex functions (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 , . The results are given in Table 2.
Euler polynomials are polynomials of degree . Thus, has zeros and has 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, 14–16].
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
S. Araci and M. Acikgoz, “A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials,” Advanced Studies in Contemporary Mathematics, vol. 22, pp. 399–406, 2012.View at: Google Scholar
R. Ayoub, “Euler zeta function,” The American Mathematical Monthly, vol. 81, pp. 1067–1086, 1974.View at: Google Scholar
A. Bayad, “Modular properties of elliptic Bernoulli and Euler functions,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 3, pp. 389–401, 2010.View at: Google Scholar
I. N. Cangul, H. Ozden, and Y. Simsek, “A new approach to q-Genocchi numbers and their interpolation functions,” Nonlinear Analysis: Theory, Methods and Applications, vol. 71, no. 12, pp. e793–e799, 2009.View at: Publisher Site | Google Scholar
L.-C. Jang, “On multiple generalized ω-Genocchi polynomials and their applications,” Mathematical Problems in Engineering, vol. 2010, Article ID 316870, 8 pages, 2010.View at: Publisher Site | Google Scholar
J. Y. Kang, H. Y. Lee, and N. S. Jung, “Some relations of the twisted q-Genocchi numbers and polynomials with weight and weak Weight ,” Abstract and Applied Analysis, vol. 2012, Article ID 860921, 9 pages, 2012.View at: Publisher Site | Google Scholar
M. S. Kim and S. Hu, “On p-adic Hurwitz-type Euler Zeta functions,” Journal of Number Theory, no. 132, pp. 2977–3015, 2012.View at: Google Scholar
T. Kim, “Barnes-type multiple q-zeta functions and q-Euler polynomials,” Journal of Physics A: Mathematical and Theoretical, vol. 43, Article ID 255201, 11 pages, 2010.View at: Google Scholar
T. Kim, “Note on the Euler q-zeta functions,” Journal of Number Theory, vol. 129, no. 7, pp. 1798–1804, 2009.View at: Publisher Site | Google Scholar
T. Kim, “On p-adic interpolating function for q-Euler numbers and its derivatives,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 598–608, 2008.View at: Publisher Site | Google Scholar
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, vol. 2005, no. 2, pp. 171–181, 2005.View at: Publisher Site | Google Scholar
H. Ozden and Y. Simsek, “A new extension of q-Euler numbers and polynomials related to their interpolation functions,” Applied Mathematics Letters, vol. 21, no. 9, pp. 934–939, 2008.View at: Publisher Site | Google Scholar
K. H. Park, S.-H. Rim, and E. J. Moon, “On Genocchi numbers and polynomials,” Abstract and Applied Analysis, vol. 2008, Article ID 898471, 7 pages, 2008.View at: Publisher Site | Google Scholar
C. S. Ryoo, “A numerical computation on the structure of the roots of q-extension of Genocchi polynomials,” Applied Mathematics Letters, vol. 21, no. 4, pp. 348–354, 2008.View at: Publisher Site | Google Scholar
C. S. Ryoo, “Calculating zeros of the second kind Euler polynomials,” Journal of Computational Analysis and Applications, vol. 12, pp. 828–833, 2010.View at: Google Scholar
C. S. Ryoo, T. Kim, and R. P. Agarwal, “A numerical investigation of the roots of q-polynomials,” International Journal of Computer Mathematics, vol. 83, no. 2, pp. 223–234, 2006.View at: Publisher Site | Google Scholar
Y. Simsek, “Generating functions of the twisted Bernoulli Numbers and polynomials associated with their interpolation functions,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 251–257, 2008.View at: Google Scholar
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.View at: Publisher Site | Google Scholar
Y. Simsek, V. Kurt, and D. Kim, “New approach to the complete sum of products of the twisted (h, q)-Bernoulli numbers and polynomials,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 44–56, 2007.View at: Publisher Site | Google Scholar