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A NUMERICAL INVESTIGATION ON THE ZEROS OF THE TANGENT POLYNOMIALS
A NUMERICAL INVESTIGATION ON THE ZEROS OF THE TANGENT POLYNOMIALS
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(3_4): 315-322
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : April 08, 2013
  • Accepted : February 08, 2014
  • Published : September 28, 2014
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About the Authors
C. S. RYOO

Abstract
In this paper, we observe the behavior of complex roots of the tangent polynomials Tn ( x ), using numerical investigation. By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the tangent polynomials Tn ( x ). Finally, we give a table for the solutions of the tangent polynomials Tn ( x ). AMS Mathematics Subject Classification : 11B68, 11S40, 11S80.
Keywords
1. Introduction
In the 21st century, the computing environment would make more and more rapid progress. Numerical experiments of Bernoulli polynomials, Euler polynomials, Genocchi polynomials, and tangent polynomials have been the subject of extensive study in recent year and much progress have been made both mathematically and computationally(see [1 - 15] ). Using computer, a realistic study for tangent polynomials Tn ( x ) is very interesting. It is the aim of this paper to observe an interesting phenomenon of ‘scattering’ of the zeros of the tangent polynomials Tn ( x ) in complex plane. Throughout this paper, we always make use of the following notations: ℕ denotes the set of natural numbers, ℝ denotes the set of real numbers, and ℂ denotes the set of complex numbers. Tangent numbers was introduced in [6] . First, we introduce the tangent numbers and tangent polynomials. As well known definition, the tangent numbers Tn (cf. [6] ) are defined by
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Here is the list of the first tangent’s numbers:
  • T0= 1,
  • T1= -1,
  • T3= 2,
  • T5= -16,
  • T7= 272,
  • T9= -7936,
  • T11= 353792,
  • T13= -22368256,
  • T15= 1903757312,
  • T17= -209865342976,
  • T19= 29088885112832,
  • T21= -4951498053124096,
  • T23= 1015423886506852352,
  • T25= -246921480190207983616,
  • T27= 70251601603943959887872,
  • T29= -23119184187809597841473536,
  • T31= 8713962757125169296170811392.
In [6] , we introduced the tangent polynomials Tn ( x ). The tangent polynomials Tn ( x ) are defined by the generating function:
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where we use the technique method notation by replacing T ( x ) n by Tn ( x ) symbolically. Note that
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In the special case x = 0, we define Tn (0) = Tn .
Because
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it follows the important relation
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Since
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we see that
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Since Tn (0) = Tn , by (1.2), we have the following theorem.
Theorem 1.1. For n ∈ ℕ, we have
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Then, it is easy to deduce that Tk ( x ) are polynomials of degree k . Here is the list of the first tangent’s polynomials:
  • T0(x) = 1,
  • T1(x) =x-1,
  • T2(x) =x2-2x,
  • T3(x) =x3-3x2+2,
  • T4(x) =x4-4x3+8x,
  • T5(x) =x5-5x4+20x2-16,
  • T6(x) =x6-6x5+40x3-96x,
  • T7(x) =x7-7x6+70x4-336x2+272,
  • T8(x) =x8-8x7+112x5-896x3+2176x,
  • T9(x) =x9-9x8+168x6-2016x4+9792x2-7936,
  • T10(x) =x10-10x9+240x7-4032x5+32640x3-79360x,
2. Beautiful zeros of the tangent polynomials
In this section, we display the shapes of the tangent polynomials Tn ( x ) and we investigate the zeros of the tangent polynomials Tn ( x ). For n = 1, · · · , 10, we can draw a plot of Tn ( x ), respectively. This shows the ten plots combined into one. We display the shape of Tn ( x ),−7 ≤ x ≤ 7( Figure 1 ). Next, we investigate the beautiful zeros of the Tn ( x ) by using a computer. We plot the zeros of Tn ( x ) for n = 20, 30, 40, 60, and x ∈ ℂ( Figure 2 ). Stacks of zeros of Tn ( x ) for 1 ≤ n ≤ 50 from a 3-D structure are presented( Figure 3 ). In Figure 2 (top-left), we choose n = 20, In Figure 2 (top-right), we choose n = 30, In Figure 2 (bottom-left), we choose n = 40, In Figure 2 (bottom-right), we choose n = 50.
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Curve of tangent polynomials Tn(x)
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Zeros of Tn(x) for n = 20; 30; 40; 50
Our numerical results for approximate solutions of real zeros of Tn ( x ) are displayed in Table 1 . The results are obtained by Mathematica software.
Numbers of real and complex zeros ofTn(x)
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Numbers of real and complex zeros of Tn(x)
We observe a remarkably regular structure of the complex roots of tangent polynomials. We hope to verify a remarkably regular structure of the complex roots of tangent polynomials( Table 1 ).
Next, we calculated an approximate solution satisfying Tn ( x ) and x ∈ ℝ. The results are given in Table 2 .
Approximate solutions ofTn(x) = 0,x∈ ℝ
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Approximate solutions of Tn(x) = 0, x ∈ ℝ
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Stacks of zeros of Tn(x), 1 ≤ n ≤ 50
We plot the real zeros of the tangent polynomials Tn ( x ) for x ∈ ℂ ( Figure 4 ).
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Real zeros of Tn(x), 1 ≤ n ≤ 50
3. Observations
Since
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we have the following theorem.
Theorem 3.1. For any positive integer n, we have
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From (1.1), we have
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Comparing the coefficient of
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on both sides of (3.2), we get the following theorem.
Theorem 3.2. For any positive integer n, we have
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By (3.3), we have the following corollary.
Corollary 3.3. For n ∈ ℕ, we have
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The question is: what happens with the reflexive symmetry (3.1), when one considers tangent polynomials? Prove that Tn ( x ), x ∈ ℂ has Re ( x ) = 1 reflection symmetry in addition to the usual Im ( x ) = 0 reflection symmetry analytic complex functions( Figures 2 - 4 ). Prove that Tn ( x ) = 0 has n distinct solutions. Find the numbers of complex zeros
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of Tn ( x ), Im ( x )≠ 0: Since n is the degree of the polynomial Tn ( x ), the number of real zeros
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lying on the real plane Im ( x ) = 0 is then
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= n
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, where
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denotes complex zeros. See Table 1 for tabulated values of
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and
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. More studies and results in this subject we may see references [8] , [9] , [13] , [14] .
BIO
C. S. Ryoo received Ph.D. degree from Kyushu University. His research interests focus on the numerical verifcation method, scientifc computing and p-adic functional analysis.
Department of Mathematics, Hannam University, Daejeon, 306-791, Korea
e-mail: ryoocs@hnu.kr
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