Research Article  Open Access
C. S. Ryoo, T. Kim, "A Note on the Extension of Bernoulli Numbers and Bernoulli Polynomials", Discrete Dynamics in Nature and Society, vol. 2010, Article ID 807176, 11 pages, 2010. https://doi.org/10.1155/2010/807176
A Note on the Extension of Bernoulli Numbers and Bernoulli Polynomials
Abstract
We observe the behavior of roots of the ()extension of Bernoulli polynomials . By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the qextension of Bernoulli polynomials . The main purpose of this paper is also to investigate the zeros of the ()extension of Bernoulli polynomials . Furthermore, we give a table for the zeros of the ()extension of Bernoulli polynomials .
1. Introduction
Throughout this paper , and will be denoted by the ring of rational integers, the ring of adic integers, the field of adic rational numbers, and the completion of algebraic closure of , respectively, compare with [1–6]. Let be the normalized exponential valuation of with When one talks of extension, is variously considered as an indeterminate, a complex number or adic number If then we normally assume that so that for If , then we normally assume that For the adic integral (or Volkenborn integration) was defined as where compare with [1–8]. Thus, we note that By (1.2), we easily see that where
In (1.3), if we take then we have for .
Recently, many mathematicians have studied Bernoulli numbers and Bernoulli polynomials. Bernoulli polynomials possess many interesting properties and arising in many areas of mathematics and physics. For more studies in this subject we may see references [1–8]. The motivation for this study comes from the following papers. Some interesting analogues of the Bernoulli numbers and polynomials were investigated by Ryoo and Kim [6]. We begin by recalling here definitions of extension of Bernoulli numbers and polynomials as follows.
Definition 1.1 (see [6]). The extension of Bernoulli numbers and polynomials is defined by means of the generating functions as follows:
Note that , and where are the th Bernoulli numbers.
By (1.4) and (1.5), we have the following Witt formula. For with , we have In this paper, we investigate the extension of Bernoulli numbers and Bernoulli polynomials in order to obtain some interesting results and explicit relationships. The aim of this paper to observe an interesting phenomenon of “scattering” of the zeros of the extension of Bernoulli polynomials . The outline of this paper is as follows. In Section 2, we study the extension of Bernoulli polynomials . In Section 3, we describe the beautiful zeros of the extension of Bernoulli polynomials using a numerical investigation. Also we display distribution and structure of the zeros of the extension of Bernoulli polynomials by using computer. By using the results of our paper, the readers can observe the regular behaviour of the roots of the extension of Bernoulli polynomials . Finally, we carried out computer experiments for demonstrating a remarkably regular structure of the complex roots of the extension of Bernoulli polynomials .
2. Basic Properties for the Extension of Bernoulli Numbers and Bernoulli Polynomials
Let be a complex number with and . By the meaning of (1.5), the extension of Bernoulli numbers and Bernoulli polynomials is defined by means of the following generating function: respectively.
Here is the list of the first extension of Bernoulli numbers .
because it follows the important relation We have the integral formula as follows:
Here is the list of the first extension of Bernoulli Polynomials .
Since we have the following theorem.
Theorem 2.1. extension of Bernoulli polynomials satisfies the following relation:
From (2.2), we can derive the following equality:
Hence, we obtain the following difference equation.
Theorem 2.2. For any positive integer , we obtain
3. Distribution and Structure of the Zeros
In this section, we assume that , with . We observed the behavior of real roots of the extension of Bernoulli polynomials . We display the shapes of the extension of Bernoulli polynomials and we investigate the zeros of the extension of Bernoulli polynomials . For , we can draw a plot of the extension of Bernoulli polynomials , respectively. This shows the ten plots combined into one. We display the shape of (Figure 1). We investigate the beautiful zeros of the extension of Bernoulli polynomials by using a computer. We plot the zeros of the extension of Bernoulli polynomials for and (Figure 2).
(a)
(b)
(c)
(d)
Our numerical results for approximate solutions of real zeros of are displayed (Tables 1 and 2).


We plot the zeros of extension of Bernoulli polynomials for , and (Figure 3). We plot the zeros of extension of Bernoulli polynomials for , and (Figure 4).
(a)
(b)
(c)
(d)
(a)
(b)
We observe a remarkably regular structure of the complex roots of the extension of Bernoulli polynomials . We hope to verify a remarkably regular structure of the complex roots of the extension of Bernoulli polynomials (Table 1). This numerical investigation is especially exciting because we can obtain an interesting phenomenon of scattering of the zeros of the extension of Bernoulli polynomials . These results are used not only in pure mathematics and applied mathematics, but also used in mathematical physics and other areas. Next, we calculated an approximate solution satisfying the extension of Bernoulli polynomials . The results are given in Table 2.
Stacks of zeros of for from a 3D structure are presented (in Figure 5).
Figure 6 presents the distribution of real zeros of the extension of Bernoulli polynomials for .
Figure 7 presents the distribution of real zeros of the extension of Bernoulli polynomials for .
Figure 8 presents the distribution of real zeros of the Bernoulli polynomials for .
4. Direction for Further Research
In [7], we observed the behavior of complex roots of the Bernoulli polynomials , using numerical investigation. Prove that , has reflection symmetry in addition to the usual reflection symmetry analytic complex functions. The obvious corollary is that the zeros of will also inherit these symmetries. where denotes complex conjugation (see [7]). Finally, we shall consider the more general problems. Prove that has distinct solutions. If has and reflection symmetries, and nondegenerate zeros, then of the distinct zeros will satisfy (4.1). If the remaining one zero is to satisfy (4.1) too, it must reflect into itself, and therefore it must lie at , the center of the structure of the zeros, that is, Prove that has distinct solutions, that is, all the zeros are nondegenerate. 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 . Find the equation of envelope curves bounding the real zeros lying on the plane. We prove that has reflection symmetry analytic complex functions. If , then , where denotes complex conjugate (see Figures 2, 3, and 4). Observe that the structure of the zeros of the Bernoulli polynomials resembles the structure of the zeros of the Bernoulli polynomials as (see Figures 3, 4, and 5). In order to study the extension of Bernoulli polynomials , we must understand the structure of the extension of Bernoulli polynomials . Therefore, using computer, a realistic study for the extension of Bernoulli polynomials plays an important part. The author has no doubt that investigation along this line will lead to a new approach employing numerical method in the field of research of the extension of Bernoulli polynomials to appear in mathematics and physics. For related topics, the interested reader is referred to [3–8].
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Copyright
Copyright © 2010 C. S. Ryoo and T. Kim. 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.