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

In this paper, we study the analytic continuation Tq(s) and Tq(s,w) of the q-Tangent numbers Tn,q and q-Tangent polynomials Tn,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 Tq(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.