Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences

Let Pn and Qn be the polynomials obtained by repeated differentiation of the tangent and secant functions respectively. From the exponential generating functions of these polynomials we develop relations among their values, which are then applied to various numerical sequences which occur as values of the Pn and Qn. For example, Pn(0) and Qn(0) are respectively the nth tangent and secant numbers, while Pn(0) +Qn(0) is the nth André number. The André numbers, along with the numbers Qn(1) and Pn(1) − Qn(1), are the Springer numbers of root systems of types An, Bn, and Dn respectively, or alternatively (following V. I. Arnol’d) count the number of “snakes” of these types. We prove this for the latter two cases using combinatorial arguments. We relate the values of Pn and Qn at √ 3 to certain “generalized Euler and class numbers” of D. Shanks, which have a combinatorial interpretation in terms of 3-signed permutations as defined by R. Ehrenborg and M. A. Readdy. Finally, we express the values of Euler polynomials at any rational argument in terms of Pn and Qn, and from this deduce formulas for Springer and Shanks numbers in terms of Euler polynomials.