International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 817157, 14 pages

doi:10.1155/2012/817157

Research Article

Generalized -Euler Numbers and Polynomials Associated with -Adic -Integral on

H. Y. Lee, N. S. Jung, J. Y. Kang, and C. S. Ryoo

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

Received 18 June 2012; Accepted 4 October 2012

Academic Editor: A. Bayad

Copyright © 2012 H. Y. Lee et al. 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 generalize the Euler numbers and polynomials by the generalized -Euler numbers and polynomials . We observe an interesting phenomenon of “scattering” of the zeros of the generalized -Euler polynomials in complex plane.

1. Introduction

Recently, many mathematicians have studied in the area of the Euler numbers and polynomials (see [1–15]). The Euler numbers and polynomials possess many interesting properties and arising in many areas of mathematics and physics. 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 “scattering” of the zeros of 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 assume that . If , we normally assume that so that for

Compared with [1, 4, 5]. Hence, for any with in the present -adic case. Let be a fixed integer, and let be a fixed prime number. For any positive integer , we set

where lies in . For any positive integer ,

is known to be a distribution on , compared with [1–10, 14]. For

Kim defined the fermionic -adic -integral on

From (1.5), we also obtain

where (see [1–3]).

From (1.6), we obtain

where .

As 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. [1–15]).

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 . Especially, distribution of roots for is different from s. 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 (1.6) and using -adic -integral on , we have

Hence, by (2.1), we obtain

By (1.6), (2.2) and , we have

After some elementary calculations, we obtain

From (2.6), we have

with the usual convention of replacing by .

3. Basic Properties for the Generalized -Euler Numbers and Polynomials

By (2.5), we have

By (3.1), we have the following differential relation.

Theorem 3.1. For positive integers n, one has

By Theorem 3.1, we easily obtain the following corollary.

Corollary 3.2 (integral formula). Consider that

By (2.5), one obtains

By comparing coefficients of in the above equation, we arrive at the following addition theorem.

Theorem 3.3 (addition theorem). For ,

By (2.5), for , one has

By comparing coefficients of in the above equation, we arrive at the following multiplication theorem.

Theorem 3.4 (multiplication theorem). For

From (1.6), one notes that

From the above, we obtain the following theorem.

Theorem 3.5. For , we have

By (2.8) in the above, we arrive at the following corollary.

Corollary 3.6. For , one has

with the usual convention of replacing by .

From (1.7), one notes that

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

By using (4.2), one notes that

Hence, one obtains

By using the above equation, one is 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

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, and , 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).

(a)

(b)

(c)

(d)

Figure 1: Zeros of for .

(a)

(b)

(c)

(d)

Figure 2: Zeros of for .

(a)

(b)

(c)

(d)

Figure 3: Real zeros of for .

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 (top-left), we choose , and . In Figure 1 (top-right), we choose , and . In Figure 1 (bottom-left), we choose , and . In Figure 1 (bottom-right), we choose , and .

We plot the zeros of the generalized Euler polynomials for , and (Figure 2).

In Figure 2 (top-left), we choose , and . In Figure 2 (top-right), we choose , and . In Figure 2 (bottom-left), we choose , and . In Figure 2 (bottom-right), we choose and .

Plots of real zeros of for structure are presented (Figure 3).

In Figure 3 (top-left), we choose , and . In Figure 3 (top-right), we choose , and . In Figure 3 (bottom-left), we choose , and . In Figure 3 (bottom-right), we choose , and .

Stacks of zeros of for from a 3-D structure are presented (Figure 4).

Figure 4: Stacks of zeros of for .

Our numerical results for approximate solutions of real zeros of the generalized are displayed (Tables 1 and 2).

Table 1: Numbers of real and complex zeros of .

Real zeros
Complex zeros
Real zeros
Complex zeros

1
1
0
1
0

2
2
0
2
0

3
3
0
1
2

4
2
2
2
2

5
3
2
1
4

6
4
2
2
4

7
3
4
3
4

8
4
4
2
6

9
3
6
3
6

10
4
6
2
8

11
5
6
3
8

12
6
6
4
8

13
5
8
3
10

Table 2: Approximate solutions of .

1

2

3

4

5

6

7

8

9

10

11

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.

Figure 5 shows the generalized -Euler polynomials for real and , with the zero contour indicated in black (Figure 5). In Figure 5 (top-left), we choose , , and . In Figure 5 (top-right), we choose , , and . In Figure 5 (bottom-left), we choose , , and . In Figure 5 (bottom-right), we choose , and .

(a)

(b)

(c)

(d)

Figure 5: Zero contour of .

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 1–5). 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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