Polynomials related to [formula omitted]-analog of the generalized derivative polynomials

Abstract

The derivative polynomials introduced by Knuth and Buckholtz play an important role in calculating the tangent and secant numbers. In this paper, we study a class of polynomials defined by the Möbius transformation on q-analog of the generalized derivative polynomials for the secant function. In a certain sense, it can be looked on as a common q-analog of two kinds of alternating Eulerian polynomials. We derive many properties of such polynomials, such as symmetry, unimodality, strong x-log-convexity, Hurwitz stability, semi-γ-positivity and convolutional relation. Moreover, we also obtain its exponential generating function, the Jacobi continued fraction expansion of its ordinary generating function and its explicit formula.