##### Moscow Journal of Combinatorics and Number Theory, vol. 2, issue 3, 2012, pp. 34-84, [pp. 232-282].

# Dominique Foata, Guo-Niu Han

#
MULTIVARIABLE TANGENT AND SECANT q-DERIVATIVE POLYNOMIALS

**Abstract.**
The derivative polynomials introduced by Knuth and
Buckholtz in their calculations of the tangent and secant
numbers are extended to a multivariable
q-environment. The n-th q-derivatives of the
classical q-tangent tan_q(u) and of the two
q-secants sec_q(u), Sec_q(u) are given
two polynomial expressions in tan_q(u),
sec_q(u), Sec_q(u), indexed by triplets of integers for the first
class, and compositions of integers for the second. The
functional relation between those two classes is fully given
by means of combinatorial techniques. Moreover, those
polynomials are proved to be generating functions for
so-called t-permutations by multivariable statistics. By giving special values to those polynomials we recover classical $q$-polynomials such as the Carlitz q-Eulerian polynomials and the
(t,q)-tangent and -secant analogs recently introduced.
They also provide q-analogs for the Springer numbers.
Finally, the t-compositions used in this paper furnish a combinatorial interpretation to one of the Fibonacci triangles.

foata at unistra dot fr,
guoniu dot han at unistra dot fr

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