A family of polynomials interpolating several classical series of numbers

We give a common polynomial extension of the Euler numbers, Genocchi numbers, Eulerian polynomials, the recent median Euler numbers. We first study some general algebraic properties of these polynomials. which include the continued fraction expansion of its ordinary generating function and, by establishing the connection with a generating function of some staircases introduced by Dumont, we get several combinatorial interpretations of these polynomials and then several new combinatorial interpretations of the above classical numbers. Finally, we study also a similar extension of Springer numbers. (C) 1996 Academic Press, Inc.