Abstract
In this paper we construct the new analogues of Genocchi the numbers and polynomials. We also observe the behavior of complex roots of the -Genocchi polynomials , using numerical investigation. By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the -Genocchi polynomials . Finally, we give a table for the solutions of the -Genocchi polynomials .
1. Introduction
Many mathematicians have the studied Bernoulli numbers and polynomials, the Euler numbers and polynomials, and the Genocchi numbers and the Genocchi polynomials. The Bernoulli numbers and polynomials, the Euler numbers and polynomials, and the Genocchi polynomials posses many interesting properties and arising in many areas of mathematics and physics (see [1–12]). We introduce the new analogs of the Genocchi numbers and polynomials. In the 21st century, the computing environment would make more and more rapid progress. Using computer, a realistic study for new analogs of Genocchi numbers and polynomials is very interesting. It is the aim of this paper to observe an interesting phenomenon of “scattering” of the zeros of -Genocchi polynomials . The outline of this paper is as follows. In Section 2, we study the -Genocchi polynomials . In Section 3, we describe the beautiful zeros of -Genocchi polynomials using a numerical investigation. Also we display distribution and structure of the zeros of the -Genocchi polynomials by using computer. By using the results of our paper, the readers can observe the regular behaviour of the roots of -Genocchi polynomials . Finally, we carried out computer experiments that demonstrate a remarkably regular structure of the complex roots of -Genocchi polynomials . Throughout this paper we use the following notations. By we denote the ring of -adic rational integers, denotes the field of rational numbers, denotes the field of -adic rational numbers, denotes the complex number field, and denotes the completion of algebraic closure of . 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 Compare [1, 2, 4, 10, 11, 13–16]. 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 , cf. [1, 2, 4, 5, 9, 10, 13]. We say that is a uniformly differentiable function at a point and denote this property by , if the difference quotients have a limit as . For the -deformed bosonic -adic integral of the function is defined by Kim: Note that where . Now, the fermionic -adic invariant -integral on is defined as If we take in (1.7), then we easily see that From (1.8), we obtain where . First, we introduce the Genocchi numbers and the Genocchi polynomials. The Genocchi numbers are defined by the generating function: Compare [4, 9–11, 17], where we use the technique method notation by replacing by symbolically. We consider the Genocchi polynomials as follows: Note that . In the special case , we define .
2. An Analogue of the Genocchi Numbers and Polynomials
The versions of -Genocchi numbers and polynomials, which were derived from different considerations and different formulas, were defined by Kim [13, 14]. Kim [14] treated analogue of the Genocchi numbers, which is called -analogue of the Genocchi numbers. Kim defined the -extension of the Genocchi numbers and polynomials as follows: In [14], Kim introduced the -analogue of the Genocchi polynomials as follows: We now consider another construction -Genocchi numbers and polynomials. In (1.8), if we take , then one has Let us define the -Genocchi numbers and polynomials as follows: Note that , , where are the th Genocchi numbers. By (2.4) and (2.5), we obtain the following Witt's formula.
Theorem 2.1. For with , we have
By the above theorem, easily see that Let be a complex number with . By the meaning of (1.10) and (1.11), let us define the -Genocchi numbers and polynomials as follows: For -Euler numbers, Kim constructed -Euler numbers which can be uniquely determined by with the usual convention of symbolically replacing by , where denotes the -Euler numbers. For -Genocchi numbers, we have the following theorem.
Theorem 2.2. -Genocchi numbers are defined inductively by with the usual convention about replacing by in the binomial expansion.
Proof. From (2.4), we obtain
which yields
Using the Taylor expansion of exponential function, we have
The result follows by comparing the coefficients.
Here is the list of the first -Genocchi numbers :
We display the shapes of the -Genocchi numbers . For , we can draw a curve of , respectively. This shows the ten curves combined into one. We display the shape of (Figure 1).
Because
it follows the important relation
Here is the list of the first the -Genocchi polynomials :
Since
we have the following theorem.
Theorem 2.3. -Genocchi polynomials satisfy the following relation: It is easy to see that Hence we have the following theorem.
Theorem 2.4. For any positive integer (=odd), one obtains
3. Distribution and Structure of the Zeros
In this section, we investigate the zeros of the -Genocchi polynomials by using computer. We display the shapes of the -Genocchi polynomials . For , we can draw a curve of , respectively. This shows the ten curves combined into one. We display the shape of (Figures 2, 3, 4, and 5).
We plot the zeros of for (Figures 6, 7, 8, and 9).
Next, we plot the zeros of for . (Figures 10, 11, 12, and 13).
In Figures 6, 7, 8, 9, 10, 11, 12, and 13, , has reflection symmetry. This translates to the following open problem: prove or disprove: , has reflection symmetry. Our numerical results for numbers of real and complex zeros of , are displayed in Table 1.
Figure 15 shows the distribution of real zeros of for .
In Figure 15(a), we choose . In Figure 15(b), we choose . In Figure 15(c), we choose . In Figure 15(d), we choose .
We calculated an approximate solution satisfying . The results are given in Tables 2 and 3.
The plot above shows for real and , with the zero contour indicated in black (Figure 16). In Figure 16(a), we choose . In Figure 16(b), we choose . In Figure 16(c), we choose . In Figure 16(d), we choose .
We will consider the more general open problem. In general, how many roots does have? Prove or disprove: has distinct solutions. Find the numbers of complex zeros of . Prove or give a counterexample: Conjecture: 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 and the equation of a trajectory curve running through the complex zeros on any one of the arcs. For , we can draw a plot of the , respectively. This shows the ten curves combined into one. These figures give mathematicians an unbounded capacity to create visual mathematical investigations of the behavior of the and roots of the (Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16). Moreover, it is possible to create new mathematical ideas and analyze them in ways that generally are not possible by hand. 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 -Genocchi polynomials to appear in mathematics and physics. For related topics the interested reader is referred to [15–19].
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