Abstract
In this paper, 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 related to solution of Bernoulli equation. Finally, we observe an interesting phenomenon of “scattering” of the zeros of .
1. Introduction
By using software, many 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. 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. Mathematicians have studied different kinds of the Euler, Bernoulli, Tangent, and Genocchi numbers and polynomials. Numerical experiments of Bernoulli polynomials, Euler polynomials, Genocchi polynomials, and Tangent polynomials have been the subject of extensive study in recent year and much progress has been made both mathematically and computationally (see [1–18]). 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 integers, denotes the set of real numbers, and denotes the set of complex numbers. Let be a complex number with and . Bernoulli equation is one of the well known nonlinear differential equations of the first order. It is written as where and are continuous functions. For and the equation is linear, and otherwise it is nonlinear. When , the Bernoulli equation has the solution which is the function of exponential generating function of the Euler numbers. Simsek [18] introduced the -Euler numbers and polynomials . He gave recurrence identities -Euler polynomials and the alternating sums of powers of consecutive -integers. In [13], we described the beautiful zeros of the -Euler polynomials using a numerical investigation. Also we investigated distribution and structure of the zeros of the -Euler polynomials by using computer.
Let us define the -Euler numbers and polynomials as follows: Observe that if , then and , where and denote the Euler polynomials and the numbers, respectively (see [2, 5, 8, 16, 17]).
Thus -Euler numbers are defined by means of the generating function As is well known, when a special Bernoulli equation has the solution That is, the Bernoulli equation has the solution which is the function of exponential generating function of the -Euler numbers. Thus, a realistic study for the analytic continued polynomials is very interesting by using computer. It is the aim of this paper to observe an interesting phenomenon of “scattering” of the zeros of the analytic continued polynomials in complex plane.
By using computer, the -Euler numbers can be determined explicitly. A few of them are
Theorem 1. For , we have
By Theorem 1, after some elementary calculations, we have
Since , by (9), 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 introduce the -Euler zeta function and Hurwitz -Euler zeta function. By -Euler zeta function, we consider the function as the analytic continuation of -Euler numbers. For more studies in this subject, you may see [2–5, 7–9, 12, 13, 18].
From (4), we note that By using the above equation, we are now ready to define -Euler zeta functions.
Definition 2. Let with . Consider
Observe that is a meromorphic function on . Clearly, (see [3, 4, 7–9, 12, 13, 18]). 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.
Theorem 3. For , we have
Observe that function interpolates numbers at nonnegative integers.
By using (3), we note that By (16), we are now ready to define the Hurwitz-type -Euler zeta functions.
Definition 4. Let with . Consider
Note that is a meromorphic function on (see [3, 4, 7–9, 12, 13, 18]). Relation between and is given by the following theorem.
Theorem 5. For , we have
We now consider the function as the analytic continuation of -Euler numbers. From the above analytic continuation of -Euler numbers, we consider
In Figure 1(a), we choose and . In Figure 1(b), we choose and .
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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 , as follows: From the relation (21), we can define the other analytic continued half of -Euler numbers By (22), we have The curve runs through the points and grows asymptotically as (see Figure 2).
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In Figure 2(a), we choose and . In Figure 2(b), we choose and .
3. Analytic Continuation of Euler Polynomials
In this section, we observe the analytic continued -Euler polynomials. Looking back at (13) and (22), 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 (25), we obtain analytic continuation of -Euler polynomials for and as follows: By using (26), 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(b), we draw and axes but no axis in three dimensions. In Figure 4(c), we draw and axes but no axis in three dimensions. In Figure 4(d), we draw and axes but no axis in three dimensions.
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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: 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).
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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
Observe that , , has reflection symmetry in addition to the usual reflection symmetry analytic complex functions (see [14]). The question is, what happens with the reflexive symmetry (29), when one considers -Euler polynomials? Prove that , , has no reflection symmetry analytic complex functions (Figure 4). However, we observe that , , has reflection symmetry analytic complex functions (Figure 5).
Stacks of zeros of for , , from a 3-D structure are presented (Figure 6).
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In Figure 6(b), we draw and axes but no axis in three dimensions. In Figure 6(c), we draw and axes but no axis in three dimensions. In Figure 6(d), we draw and axes but no axis in three dimensions.
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.
In Figure 7, we plot the real zeros of the -Euler polynomials for , , , and (Figure 7). In Figure 7(a), we choose . In Figure 7(b), we choose . We want to find a formula that best fits a given set of data points. The least squares method is used to fit polynomials or a set of functions to a given set of data points. Using the least squares method, we can find and such that is the least squares fit to the data given in Table 2. The graph of the data points is shown in Figure 7. We obtain for . We also obtain for and for . The real zero asymptotically as .
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The -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 in this subject, you may see [6, 11–15].
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.