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Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Orthogonal polynomials represented by CW -spheres. | Orthogonal polynomials represented by CW-spheres Gábor Hetyei Mathematics Department UNC Charlotte Charlotte NC 28223 Submitted Apr 2 2004 Accepted Jul 20 2004 Published Aug 16 2004 Abstract Given a sequence Qn x n 0 of symmetric orthogonal polynomials defined by a recurrence formula Qn x Vn x Qn-i x vn 1 Qn-2 x with integer Vi s satisfying Vi 2 we construct a sequence of nested Eulerian posets whose ce-index is a non-commutative generalization of these polynomials. Using spherical shellings and direct calculations of the cd-coefficients of the associated Eulerian posets we obtain two new proofs for a bound on the true interval of orthogonality of Qn x n 0. Either argument can replace the use of the theory of chain sequences. Our cd-index calculations allow us to represent the orthogonal polynomials as an explicit positive combination of terms of the form xn-2r x2 1 r. Both proofs may be extended to the case when the Vi s are not integers and the second proof is still valid when only Vi 1 is required. The construction provides a new limited testing ground for Stanley s non-negativity conjecture for Gorenstein posets and suggests the existence of strong links between the theory of orthogonal polynomials and flag-enumeration in Eulerian posets. Introduction In a recent paper 13 the present author constructed a sequence of nested Eulerian partially ordered sets whose ce-index generalizes the Tchebyshev polynomials of the first On leave from the Renyi Mathematical Institute of the Hungarian Academy of Sciences. 2000 Mathematics Subject Classification Primary 05E35 Secondary 06A07 57Q15 Key words and phrases partially ordered set Eulerian flag orthogonal polynomial. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2 2004 R4 1 kind. The main goal of that paper was to propose a new class of posets to test Stanley s non-negativity conjecture 17 Conjecture 2.1 on the cd-index of Gorenstein posets. In this paper we construct a similar sequence Q0 Q1 . of nested Eulerian posets for
