Dynamics of the zeros of analytic continued polynomials and differential equations associated with \(q\)tangent polynomials
Authors
Cheon Seoung Ryoo
 Department of Mathematics, Hannam University, Daejeon 306791, Republic of Korea.
Kyung Won Hwang
 Department of Mathematics, DongA University, Busan 604714, Republic of Korea.
Do Jin Kim
 Department of Mathematics, Kyungpook National University, Daegu, 702701, Republic of Korea.
Nam Soon Jung
 College of Talmage Liberal Arts, Hannam University,, Daejeon 306791, Republic of Korea.
Abstract
In this paper, we study the analytic continuation \(T_q(s)\) and \(T_q(s,w)\) of the \(q\)Tangent numbers \(T_{n, q}\) and \(q\)Tangent polynomials \(T_{n, q}(x)\) introduced by authors.
The new concept of dynamics of the zeros of
analytic continued \(q\)tangent polynomials is investigated observing an
interesting phenomenon of `scattering' of the zeros of \(T_q(s,
w)\). Finally, we study linear differential equations arising from the generating functions of \(q\)tangent polynomials giving explicit identities for the \(q\)tangent polynomials.
Share and Cite
ISRP Style
Cheon Seoung Ryoo, Kyung Won Hwang, Do Jin Kim, Nam Soon Jung, Dynamics of the zeros of analytic continued polynomials and differential equations associated with \(q\)tangent polynomials, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 6, 785797
AMA Style
Ryoo Cheon Seoung, Hwang Kyung Won, Kim Do Jin, Jung Nam Soon, Dynamics of the zeros of analytic continued polynomials and differential equations associated with \(q\)tangent polynomials. J. Nonlinear Sci. Appl. (2018); 11(6):785797
Chicago/Turabian Style
Ryoo, Cheon Seoung, Hwang, Kyung Won, Kim, Do Jin, Jung, Nam Soon. "Dynamics of the zeros of analytic continued polynomials and differential equations associated with \(q\)tangent polynomials." Journal of Nonlinear Sciences and Applications, 11, no. 6 (2018): 785797
Keywords
 Tangent numbers and polynomials
 \(q\)tangent polynomial
 \(q\)tangent Zeta function
 analytic continuation
 analytic continued \(q\)tangent polynomials
 zeros
 differential equations
MSC
References

[1]
R. Ayoub, Euler and the zeta function, Amer. Math., Monthly, 81 (1974), 1067–1086.

[2]
A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions, Vol. III., Krieger Publishing Co., Melbourne (1981)

[3]
J. Y. Kang, H. Y. Lee, N. S. Jung , Some relations of the twisted qGenocchi numbers and polynomials with weight \(\alpha\) and weak Weight \(\beta\), Abstr. Appl. Anal., 2012 (2012), 9 pages.

[4]
T. Kim , Euler numbers and polynomials associated with zeta functions, Abstr. Appl. Anal., 2008 (2008), 11 pages.

[5]
M.S. Kim, S. Hu , On padic Hurwitztype Euler Zeta functions, J. Number Theory, 132 (2012), 2977–3015.

[6]
T. Kim, D. S. Kim, C. S. Ryoo, H. I. Kwon , Differential equations associated with Mahler and ShefferMahler polynomials, , ( submitted for publication.),

[7]
T. Kim, C. S. Ryoo, L. C. Jang, S. H. Rim, Exploring the qRiemann Zeta function and qBernoulli polynomials, Discrete Dyn. Nat. Soc., 2005 (2005), 171–181.

[8]
H. Ozden, Y. Simsek , A new extension of qEuler numbers and polynomials related to their interpolation functions, Appl. Math. Lett., 21 (2008), 934–938.

[9]
S. H. Rim, K. H. Park, E. J. Moon , On Genocchi numbers and polynomials, Abstr. Appl. Anal., 2008 (2008), 7 pages.

[10]
C. S. Ryoo, A Note on the Tangent Numbers and Polynomials, Adv. Studies Theor. Phys., 7 (2013), 447–454.

[11]
C. S. Ryoo, On the qTangent Numbers and Polynomials, Appl. Math. Sci. (Ruse), 7 (2013), 4935–4941.

[12]
C. S. Ryoo, A Numerical investigation on the zeros of the tangent polynomials, J. Appl. Math. Inform., 32 (2014), 315–322.

[13]
C. S. Ryoo, Analytic Continuation of Euler Polynomials and the Euler Zeta Function, Discrete Dyn. Nat. Soc., 2014 (2014), 6 pages.

[14]
C. S. Ryoo, Differential equations associated with tangent numbers, J. Appl. Math. Inform., 34 (2016), 487–494.

[15]
C. S. Ryoo, T. Kim, R. P. Agarwal, A numerical investigation of the roots of qpolynomials, Int. J. Comput. Math., 83 (2006), 223–234.

[16]
Y. Simsek, Twisted (h, q)Bernoulli numbers and polynomials related to twisted (h, q)zeta function and Lfunction, J. Math. Anal. Appl., 324 (2006), 790–804.

[17]
Y. Simsek, Generating functions of the twisted Bernoulli numbers and polynomials, Adv. Stud. Contemp. Math., 16 (2008), 251–278.