Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: q-Eulerian polynomials and polynomials with only real zeros. | q-Eulerian polynomials and polynomials with only real zeros Shi-Mei Ma and Yi Wang y Department of Applied Mathematics Dalian University of Technology Dalian 116024 P. R. China simons_ma@ wangyi@ Submitted Jul 1 2007 Accepted Jan 4 2008 Published Jan 21 2008 Mathematics Subject Classification 05A15 26C10 Abstract Let f and F be two polynomials satisfying F x u x f x v x f0 x . We characterize the relation between the location and multiplicity of the real zeros of f and F which generalizes and unifies many known results including the results of Brenti and Branden about the q-Eulerian polynomials. 1 Introduction Let s denote the symmetric group on the set 1 2 . ng and 0102 an 2 Sn. An excedance in is an index i such that Oị i. Let exc denote the number of excedances in . The classical Eulerian polynomials An x are defined by Ao x 1 An x X xexc 1 for n 1 2Sn and have been extensively investigated. It is well known that the classical Eulerian polynomials satisfy the recurrence relation An 1 x n 1 xAn x x 1 - x A n x see Bóna 1 p. 23 for instance . In 5 Brenti considered a q-analog of the classical Eulerian polynomials defined by Ao x q 1 An x q X xexc q x for n 1 2Sn This work was supported by the National Science Foundation of China under grant number 10771027. y Corresponding author. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R17 1 where c is the number of cycles in . The first few of the q-Eulerian polynomials are Ao x q 1 Ai x q q Ạỉ x q q x q Aa x q q x2 3q 1 x q2 . Clearly An x xAn x 1 for n 1. Brenti obtained the recurrence relation An i x q nx q An x q x 1 - x -dAn x q 1 dx 5 Proposition and showed that An x q has only real nonpositive simple zeros when q is a positive rational number 5 Theorem . He also proposed the following. Conjecture 1 5 Conjecture . Let n t 2 N. Then An x t has only real zeros. The conjecture has been settled recently by Brandén 3 . Let En x q 1 x nAn 1 - q . Then it is clear that An x q has only .
