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

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.