Narayana numbers and Schur-Szego composition

JOURNAL OF APPROXIMATION THEORY

卷:161

Article

Kostov, Vladimir P.²  Martinez-Finkelshtein, Andrei³  Shapiro, Boris Z.¹ 

[1] Stockholm Univ, Dept Math, SE-10691 Stockholm, Sweden

[2] Univ Nice, Math Lab, F-06108 Nice 2, France

[3] Univ Almeria, Dept Estadist & Matemat Aplicada, Almeria 04120, Spain

关键词: Schur-Szego composition;    Composition factor;    Hyperbolic polynomial;    Self-reciprocal polynomial;    Reverted polynomial;   

DOI  :  10.1016/j.jat.2008.10.013

来源: Elsevier

【 摘 要 】

In the present paper we find a new interpretation of Narayana polynomials N-n(x) which are the generating polynomials for the Narayana numbers N-n,N-k = 1/nC(n)(k-1)C(n)(k) where C-j(i) stands for the usual binomial coefficient, i.e. C-j(i) = j!/i!(j-i)!. They count Dyck paths of length n and with exactly k peaks, see e.g. [R.A. Sulanke, The Narayana distribution, in: Lattice Path Combinatorics and Applications (Vienna, 1998), J. Statist. Plann. Inference 101 (1-2) (2002) 311-326 (special issue)] and they appeared recently in a number of different combinatorial situations, see for e.g. [. Doslic, D. Syrtan, D. Veljan, Enumerative aspects of secondary structures, Discrete Math. 285 (2004) 67-82; A. Sapounakis, I. Tasoulas, P. Tsikouras, Counting strings in Dyck paths, Discrete Math. 307 (2007) 2909-2924; F. Yano, H. Yoshida, Some set partitions statistics in non-crossing partitions and generating functions, Discrete Math. 307 (2007) 3147-3160]. Strangely enough Narayana polynomials also occur as limits as n -> infinity of the sequences of eigenpolynomials of the Schur-Szego composition map sending (n - 1)-tuples of polynomials of the form (x +1)(n-1) (x + a) to their Schur-Szego product, see below. We present below a relation between Narayana polynomials and the classical Gegenbauer polynomials which implies, in particular, an explicit formula for the density and the distribution function of the asymptotic root-counting measure of the polynomial sequence {N-n(x)}. (C) 2008 Elsevier Inc. All rights reserved.

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