Dynamics of the zeros of analytic continued polynomials and differential equations associated with \(q\)-tangent polynomials

Volume 11, Issue 6, pp 785--797 http://dx.doi.org/10.22436/jnsa.011.06.06
Publication Date: April 25, 2018 Submission Date: June 28, 2017 Revision Date: January 16, 2017 Accteptance Date: March 08, 2018


Cheon Seoung Ryoo - Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea. Kyung Won Hwang - Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea. Do Jin Kim - Department of Mathematics, Kyungpook National University, Daegu, 702-701, Republic of Korea. Nam Soon Jung - College of Talmage Liberal Arts, Hannam University,, Daejeon 306-791, Republic of Korea.


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.