The qq-tangent and qq-secant numbers via continued fractions

It is well known that the (−1)(−1)-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). Recently, two distinct qq-analogues of the latter result have been discovered by Foata and Han, and Josuat-Vergès, respectively. In this paper, we will prove some general continued fraction expansion formulae, which permit us to give a unified treatment of Josuat-Vergès’ two formulae and also to derive a new qq-analogue of the aforementioned formulae. Our approach is based on a (p,q)(p,q)-analogue of tangent and secant numbers via continued fractions and also the generating function of permutations with respect to the quintuple statistic consisting of fixed point number, weak excedance number, crossing number, nesting number and inversion number. We also give a combinatorial proof of Josuat-Vergès’ formulae by using a new linear model of derangements.