Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences

Abstract

Let $P_n$ and $Q_n$ 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 $P_n$ and $Q_n$. For example, $P_n(0)$ and $Q_n(0)$ are respectively the $n$th tangent and secant numbers, while $P_n(0)+Q_n(0)$ is the $n$th André number. The André numbers, along with the numbers $Q_n(1)$ and $P_n(1)-Q_n(1)$, are the Springer numbers of root systems of types $A_n$, $B_n$, and $D_n$ 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 $P_n$ and $Q_n$ at $\sqrt3$ 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 $P_n$ and $Q_n$, and from this deduce formulas for Springer and Shanks numbers in terms of Euler polynomials.