tailieunhanh - Báo cáo toán học: "A triple lacunary generating function for Hermite polynomials"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A triple lacunary generating function for Hermite polynomials. | A triple lacunary generating function for Hermite polynomials Ira M. Gessel Department of Mathematics Brandeis University Waltham MA USA gessel@ Pallavi Jayawant Department of Mathematics Bates College Lewiston ME USA jayawant@ Submitted Feb 26 2004 Accepted Jun 14 2005 Published Jun 19 2005 Mathematics Subject Classifications 05A15 05A19 05A40 33C45 Abstract Some of the classical orthogonal polynomials such as Hermite Laguerre Charlier etc. have been shown to be the generating polynomials for certain combinatorial objects. These combinatorial interpretations are used to prove new identities and generating functions involving these polynomials. In this paper we apply Foata s approach to generating functions for the Hermite polynomials to obtain a triple lacunary generating function. We define renormalized Hermite polynomials hn u by X hn u zn euz z2 . n n 0 and give a combinatorial proof of the following generating function 1 n Y h3n u n e w-u 3u-w 6 1 6n z2n p1 6wz 23n 3n 1 6wz 3n 2n n 0 where w 1 p1 12uz 6z uC 3uz and C x 1 p1 4x 2x is the Catalan generating function. We also give an umbral proof of this generating function. 1. Introduction The Hermite polynomials Hn u may be defined by the exponential generating function X Hn u 5 e2UZ- 1 n 0 Partially supported by NSF Grant DMS-0200596 THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R30 1 In this paper we will give a combinatorial proof of an identity for Hermite polynomials. For our combinatorial interpretation it is more convenient to take a different normalization of the Hermite polynomials which makes all the coefficients positive. Therefore we work with the polynomials hn u 77Hn I - I where i 1 which have the generating 2n 2 Ỳ Ự2 J function X hn u 5 n 0 All of our formulas for hn u are easily converted into formulas for Hn u . Foata 5 gave a combinatorial proof of Doetsch s identity 2 giving a generating function for h2n u X h2n u 1 - 2z 1 2 exp r u z Ỵ 2 n 1 2z n 0 We will prove