tailieunhanh - Báo cáo toán học: "A combinatorial derivation with Schr¨der paths of a o determinant representation of Laurent biorthogonal polynomials"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A combinatorial derivation with Schr¨der paths of a o determinant representation of Laurent biorthogonal polynomials. | A combinatorial derivation with Schroder paths of a determinant representation of Laurent biorthogonal polynomials Shuhei Kamioka Department of Applied Mathematics and Physics Graduate School of Informatics Kyoto University Kyoto 606-8501 Japan kamioka@ Submitted Aug 28 2007 Accepted May 26 2008 Published May 31 2008 Mathematics Subject Classifications 05A15 42C05 05E35 Abstract A combinatorial proof in terms of Schroder paths and other weighted plane paths is given for a determinant representation of Laurent biorthogonal polynomials LBPs and that of coefficients of their three-term recurrence equation. In this process it is clarffied that Toeplitz determinants of the moments of LBPs and their minors can be evaluated by enumerating certain kinds of configurations of Schroder paths in a plane. 1 Introduction Laurent biorthogonal polynomials LBPs appeared in problems related to Thron type continued fractions T-fractions two-point Padé approximants and moment problems see . 6 and are studied by many authors . 6 4 5 11 10 . We recall fundamental properties of LBPs. Notation remark. In this paper the symbols i j k K m n and are used for nonnegative integers and for integers respectively. The symbol Xa b Z with multiple subscripts if specifically undefined denotes Xa Xb . .and Xz Let K be a field. We call a sequence Pn z 10 a sequence of Laurent biorthogonal polynomials with respect to a linear functional L K z-1 z K if for each n 0 Pn z 2 K z is a polynomial of degree n which possesses the orthogonality property L z- Pn U 0 0 z - n 11 L n 0 n. JSPS Research Fellow. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R76 1 In this paper we normalize the 0-th polynomial as Po z 1 for simplicity. The LBPs Pn z satisfy a three-term recurrence equation of the form Pn 1 z a z - 7n Pn z - nzP -1 z n 1 1 with Po z 1 and P1 z aoz 7o where the coefficients an n and 7n are some nonzero constants. The linear functional L is characterized by its moments p L z 2