کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4607786 1337884 2009 13 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Narayana numbers and Schur–Szegő composition
کلمات کلیدی
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات آنالیز ریاضی
پیش نمایش صفحه اول مقاله
Narayana numbers and Schur–Szegő composition
چکیده انگلیسی

In the present paper we find a new interpretation of Narayana polynomials Nn(x)Nn(x) which are the generating polynomials for the Narayana numbers Nn,k=1nCnk−1Cnk where Cji stands for the usual binomial coefficient, i.e. Cji=j!i!(j−i)!. They count Dyck paths of length nn and with exactly kk 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. [T. 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→∞n→∞ of the sequences of eigenpolynomials of the Schur–Szegő composition map sending (n−1)(n−1)-tuples of polynomials of the form (x+1)n−1(x+a)(x+1)n−1(x+a) to their Schur–Szegő 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 {Nn(x)}{Nn(x)}.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Approximation Theory - Volume 161, Issue 2, December 2009, Pages 464–476
نویسندگان
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