Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: The (t,q)-Analogs of Secant and Tangent Numbers. | The t q -Analogs of Secant and Tangent Numbers Dominique Foata Institut Lothaire 1 rue Murner F-67000 Strasbourg France foata@ Guo-Niu Han . Universite de Strasbourg et CNRS 7 rue Rene-Descartes F-67084 Strasbourg France Submitted Aug 6 2010 Accepted May 2 2011 Published May 13 2011 To Doron Zeilberger with our warmest regards on the occasion of his sixtieth birthday. Abstract. The secant and tangent numbers are given t q -analogs with an explicit combinatorial interpretation. This extends both analytically and combinatorially the classical evaluations of the Eulerian and Roselle polynomials at t 1. 1. Introduction As is well-known see . Ni23 p. 177-178 Co74 p. 258-259 the coefficients T2n 1 of the Taylor expansion of tanu namely u2n 1 1-1 tanu Z 2n 1 T 3 5 7 9 II u 1 T 2 16 272 7936 353792 1 3 5 7 9 11 are positive integral coefficients usually called tangent numbers while the secant numbers E2n also positive and integral make their appearances in the Taylor expansion of sec u 1 u2n 12 sec u c - . E2n n 1 v 7 1 u21 u4 5 u6 61 u81385 u84 50521 2 4 6 8 10 Key words and phrases. q-secant numbers q-tangent numbers t q -secant numbers t q -tangent numbers alternating permutations pix inverse ma jor index lec-statistic inversion number excedance number. Mathematics Subject Classifications. 05A15 05A30 33B10 THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2 2011 P7 1 On the other hand the expansion 1 s un exp Yu y2 rAAs-11 Y exp su s exp u 0 n defines a sequence An s 1 1 Y n 0 of polynomials with Positive Integral Coefficients in short PIC polynomials whose specializations An s 1 1 1 n 0 for Y 1 are called Eulerian polynomials and go back to Euler himself Eu55 while the version An s 1 1 0 n 0 for Y 0 was introduced and combinatorially interpreted by Roselle Ro68 . The two identities 1-4 A2n 1 1 1 1 0 1 nA2n 1 1 1 1 1 T2n 1 n 0 A2n 1 1 1 1 0 0 A n1 1 1 0 E2n n 0 are due to Euler Eu55 and Roselle Ro68 respectively and a joint .
