TAILIEUCHUNG - Báo cáo toán học: "The (t,q)-Analogs of Secant and Tangent Numbers"

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 .

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