In this paper, we observe the behavior of complex roots of the tangent polynomials
T_{n}
(
x
), using numerical investigation. By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the tangent polynomials
T_{n}
(
x
). Finally, we give a table for the solutions of the tangent polynomials
T_{n}
(
x
).
AMS Mathematics Subject Classification : 11B68, 11S40, 11S80.
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
T_{n}
(
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
T_{n}
(
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
T_{n}
(cf.
[6]
) are defined by
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
T_{n}
(
x
). The tangent polynomials
T_{n}
(
x
) are defined by the generating function:
where we use the technique method notation by replacing
T
(
x
)
^{n}
by
T_{n}
(
x
) symbolically. Note that
In the special case
x
= 0, we define
T_{n}
(0) =
T_{n}
.
Because
it follows the important relation
Since
we see that
Since
T_{n}
(0) =
T_{n}
, by (1.2), we have the following theorem.
Theorem 1.1.
For n
∈ ℕ,
we have
Then, it is easy to deduce that
T_{k}
(
x
) are polynomials of degree
k
. Here is the list of the first tangent’s polynomials:

T0(x) = 1,

T1(x) =x1,

T2(x) =x22x,

T3(x) =x33x2+2,

T4(x) =x44x3+8x,

T5(x) =x55x4+20x216,

T6(x) =x66x5+40x396x,

T7(x) =x77x6+70x4336x2+272,

T8(x) =x88x7+112x5896x3+2176x,

T9(x) =x99x8+168x62016x4+9792x27936,

T10(x) =x1010x9+240x74032x5+32640x379360x,
2. Beautiful zeros of the tangent polynomials
In this section, we display the shapes of the tangent polynomials
T_{n}
(
x
) and we investigate the zeros of the tangent polynomials
T_{n}
(
x
). For
n
= 1, · · · , 10, we can draw a plot of
T_{n}
(
x
), respectively. This shows the ten plots combined into one. We display the shape of
T_{n}
(
x
),−7 ≤
x
≤ 7(
Figure 1
). Next, we investigate the beautiful zeros of the
T_{n}
(
x
) by using a computer. We plot the zeros of
T_{n}
(
x
) for
n
= 20, 30, 40, 60, and
x
∈ ℂ(
Figure 2
). Stacks of zeros of
T_{n}
(
x
) for 1 ≤
n
≤ 50 from a 3D structure are presented(
Figure 3
). In
Figure 2
(topleft), we choose
n
= 20, In
Figure 2
(topright), we choose
n
= 30, In
Figure 2
(bottomleft), we choose
n
= 40, In
Figure 2
(bottomright), we choose
n
= 50.
Curve of tangent polynomials T_{n}(x)
Zeros of T_{n}(x) for n = 20; 30; 40; 50
Our numerical results for approximate solutions of real zeros of
T_{n}
(
x
) are displayed in
Table 1
. The results are obtained by Mathematica software.
Numbers of real and complex zeros ofTn(x)
Numbers of real and complex zeros of T_{n}(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
T_{n}
(
x
) and
x
∈ ℝ. The results are given in
Table 2
.
Approximate solutions ofTn(x) = 0,x∈ ℝ
Approximate solutions of T_{n}(x) = 0, x ∈ ℝ
Stacks of zeros of T_{n}(x), 1 ≤ n ≤ 50
We plot the real zeros of the tangent polynomials
T_{n}
(
x
) for
x
∈ ℂ (
Figure 4
).
Real zeros of T_{n}(x), 1 ≤ n ≤ 50
3. Observations
Since
we have the following theorem.
Theorem 3.1.
For any positive integer n, we have
From (1.1), we have
Comparing the coefficient of
on both sides of (3.2), we get the following theorem.
Theorem 3.2.
For any positive integer n, we have
By (3.3), we have the following corollary.
Corollary 3.3.
For n
∈ ℕ,
we have
The question is: what happens with the reflexive symmetry (3.1), when one considers tangent polynomials? Prove that
T_{n}
(
x
),
x
∈ ℂ has
R_{e}
(
x
) = 1 reflection symmetry in addition to the usual
Im
(
x
) = 0 reflection symmetry analytic complex functions(
Figures 2

4
). Prove that
T_{n}
(
x
) = 0 has
n
distinct solutions. Find the numbers of complex zeros
of
T_{n}
(
x
),
Im
(
x
)≠ 0: Since
n
is the degree of the polynomial
T_{n}
(
x
), the number of real zeros
lying on the real plane
Im
(
x
) = 0 is then
=
n
−
, where
denotes complex zeros. See
Table 1
for tabulated values of
and
. 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 padic functional analysis.
Department of Mathematics, Hannam University, Daejeon, 306791, Korea
email: ryoocs@hnu.kr
Comtet L.
1974
Advances combinatorics
Riedel
Dordrecht
Kim T.
(2002)
qVolkenborn integration
Russ. J. Math. Phys
9
288 
299
Kim T.
(2008)
Note on the Euler numbers and polynomials
Adv. Stud. Contemp. Math
17
131 
136
Ryoo C.S.
(2013)
A Note on the Tangent Numbers and Polynomials
Adv. Studies Theor. Phys
7
447 
454
Ryoo C.S.
(2011)
A note on the weighted qEuler numbers and polynomials
Advan. Stud. Contemp. Math
21
47 
54
Ryoo C.S.
(2008)
A numerical computation on the structure of the roots of qextension of Genocchi polynomials
Applied Mathematics Letters
21
348 
354
DOI : 10.1016/j.aml.2007.05.005
Ryoo C.S.
(2008)
Calculating zeros of the twisted Genocchi polynomials
Adv. Stud. Contemp. Math
17
147 
159
Ryoo C.S.
(2013)
On the Analogues of Tangent Numbers and Polynomials Associated with pAdic Integral on ℤp
Applied Mathematical Sciences
7
(64)
3177 
3183
Ryoo C.S.
(2013)
On the Twisted qTangent Numbers and Polynomials
Applied Mathematical Sciences
7
(99)
4935 
4941
Ryoo C. S.
(2013)
A note on the symmetric properties for the tangent polynomials
Int. Journal of Math. Analysis
7
2575 
2581
Ryoo C.S.
(2010)
A Note on the Zero of the qBernoulli polynomials
J. Appl. Math. & Informatics
28
805 
811
Ryoo C.S.
(2009)
A Note on the Reflection Symmetries of the Genocchi polynomials
J. Appl. Math. & Informatics
27
1397 
1404