Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: DERIVATIVE POLYNOMIALS, EULER POLYNOMIALS, AND ASSOCIATED INTEGER SEQUENCES. | DERIVATIVE POLYNOMIALS EULER POLYNOMIALS AND ASSOCIATED INTEGER SEQUENCES Michael E. HGFFmAN Mathematics Department . Naval Academy Annapolis MD 21402 meh@nadn. navy. mil Submitted February 16 1999 Accepted April 2 1999 Abstract. 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 Andre number. The Andre 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 a 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. 1. Introduction. Consider the sequences Pn and Qn of derivative polynomials defined by dn dn -7 tan x Pn tan x and -7 sec x Qn tan x sec x dxn dxn for integer n 0. As shown in 12 their exponential generating functions P u t X Pn u and Q u t X Qn u z n n n 0 n 0 are given by the explicit formulas 1 sin t u cos t P u t - - 7 7 cos t u sin t and Q u t ----- - -. cos t u sin t 1991 Mathematics Subject Classification. 11B83 11B68 05A15. Key words and phrases. Tangent numbers secant numbers Andre numbers Springer numbers snakes generalized Euler and class numbers Euler polynomials. Typeset by AmS-TeX 1 THE ELECTRONIC JOURNAL
