%0 Journal Article
%T Calculating Zeros of the -Genocchi Polynomials Associated with -Adic -Integral on
%A C. S. Ryoo
%J International Journal of Mathematics and Mathematical Sciences
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/210768
%X 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¨C12]). 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 ˇ°scatteringˇ± 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¨C16]. 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
%U http://www.hindawi.com/journals/ijmms/2012/210768/