We generalize the Euler numbers and polynomials by the generalized -Euler numbers and polynomials . For the complement theorem, have interesting different properties from the Euler polynomials and we observe an interesting phenomenon of βscatteringβ of the zeros of the the generalized Euler polynomials in complex plane.
1. Introduction
The Euler numbers and polynomials possess many interesting properties and arising in many areas of mathematics and physics. Recently, many mathematicians have studied in the area of the Euler numbers and polynomials (see [1β15]). In [14], we introduced that Euler equation has symmetrical roots for (see [14]). It is the aim of this paper to observe an interesting phenomenon of βscatteringβ of the zeros of the the generalized -Euler polynomials in complex plane. Throughout this paper we use the following notations. By we denote the ring of -adic rational integers, denotes the field of -adic rational numbers, denotes the completion of algebraic closure of denotes the set of natural numbers, denotes the ring of rational integers, denotes the field of rational numbers, denotes the set of complex numbers, and . Let be the normalized exponential valuation of with . When one talks of -extension, is considered in many ways such as an indeterminate, a complex number , or -adic number . If one normally assumes that . If , we normally assume that so that for .
For
Kim defined the fermionic -adic -integral on :
(cf. [5β7]).
If we take in (1.2), then we easily see that
From (1.3), we obtain
where (cf. [1β15]).
As a well-known definition, the Euler polynomials are defined by
with the usual convention of replacing by . In the special case, are called the th Euler numbers (cf. [1β15]).
Our aim in this paper is to define the generalized -Euler numbers and polynomials . We investigate some properties which are related to the generalized -Euler numbers and polynomials . In particular, distribution of roots for is different from βs. We also derive the existence of a specific interpolation function which interpolate the generalized -Euler numbers and polynomials.
2. The Generalized -Euler Numbers and Polynomials
Our primary goal of this section is to define the generalized -Euler numbers and polynomials . We also find generating functions of the generalized -Euler numbers and polynomials . Let be strictly positive real number.
The generalized -Euler numbers and polynomials are defined by
respectively.
From above definition, we obtain
Let . By (1.3) and using -adic integral on , we have
By comparing coefficients of in the above equation, we arrive at the following theorem.
Theorem 3.7. For , one has
4. The Analogue of the Euler Zeta Function
By using the generalized -Euler numbers and polynomials, the generalized -Euler zeta function and the generalized Hurwitz -Euler zeta functions are defined. These functions interpolate the generalized -Euler numbers and -Euler polynomials, respectively. Let
By applying derivative operator, to the above equation, we have
By using the above equation, we are now ready to define the generalized -Euler zeta functions.
Definition 4.1. For , one defines
Note that is a meromorphic function on . Note that if and , then which is the Hurwitz Euler zeta functions. Relation between and is given by the following theorem.
Theorem 4.2. For , one has
Observe that function interpolates numbers at nonnegative integers.
By using the above equation, we are now ready to define the generalized Hurwitz -Euler zeta functions.
Definition 4.3. Let . One defines
Note that is a meromorphic function on . Obverse that, if and , then which is the Euler zeta functions. Relation between and is given by the following theorem.
Theorem 4.4. For , one has
Observe that function interpolates numbers at nonnegative integers.
5. Zeros of the Generalized -Euler Polynomials
In this section, we investigate the reflection symmetry of the zeros of the generalized -Euler polynomials .
In the special case, are called generalized Euler polynomials . Since
we have
We observe that has reflection symmetry in addition to the usual reflection symmetry analytic complex functions.
Let
Then we have
Hence, we arrive at the following complement theorem.
Theorem 5.1 (Complement theorem). For ,
Throughout the numerical experiments, we can finally conclude that has not reflection symmetry analytic complex functions. However, we observe that has reflection symmetry (see Figures 1, 2, and 3). The obvious corollary is that the zeros of will also inherit these symmetries:
where denotes complex conjugation (see Figures 1, 2 and 3).
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(c)
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We investigate the beautiful zeros of the generalized -Euler polynomials by using a computer. We plot the zeros of the generalized Euler polynomials for , and (Figure 1).
In Figure 1(a), we choose , and . In Figure 1(b), we choose , and . In Figure 1(c), we choose , and . In Figure 1(d), we choose , and .
Plots of real zeros of for structure are presented (Figure 2).
In Figure 2(a), we choose and . In Figure 2(b), we choose and . In Figure 2(c), we choose and . In Figure 2(d), we choose and .
We investigate the beautiful zeros of the generalized by using a computer. We plot the zeros of the generalized -Euler polynomials for and (Figure 3).
In Figure 3(a), we choose and . In Figure 3(b), we choose and . In Figure 3(c), we choose and . In Figure 3(d), we choose and .
Stacks of zeros of for from a 3D structure are presented (Figure 4).
Our numerical results for approximate solutions of real zeros of the generalized are displayed (Tables 1 and 2).
We observe a remarkably regular structure of the complex roots of the generalized -Euler polynomials . We hope to verify a remarkably regular structure of the complex roots of the generalized -Euler polynomials (Table 1). Next, we calculated an approximate solution satisfying . The results are given in Table 2.
The plot above shows the generalized -Euler polynomials for real and , with the zero contour indicated in black (Figure 5). In Figure 5(a), we choose and . In Figure 5(b), we choose and . In Figure 5(c), we choose and . In Figure 5(d), we choose and .
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Finally, we will consider the more general problems. How many roots does have? This is an open problem. Prove or disprove: has distinct solutions. Find the numbers of complex zeros of . Since is the degree of the polynomial , the number of real zeros lying on the real plane is then , where denotes complex zeros. See Table 1 for tabulated values of and . We plot the zeros of , respectively (Figures 1β5). These figures give mathematicians an unbounded capacity to create visual mathematical investigations of the behavior of the roots of the . Moreover, it is possible to create a new mathematical ideas and analyze them in ways that, generally, are not possible by hand. The authors have no doubt that investigation along this line will lead to a new approach employing numerical method in the field of research of -Euler polynomials to appear in mathematics and physics.
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