Narayana numbers and Schur-Szego composition - Mathematics > Classical Analysis and ODEsReport as inadecuate




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Abstract: In the present paper we find a new interpretation of Narayana polynomialsN nx which are the generating polynomials for the Narayana numbers N {n,k}counting Dyck paths of length n and with exactly k peaks. Strangely enoughNarayana polynomials also occur as limits as n->oo of the sequences ofeigenpolynomials of the Schur-Szego composition map sending n-1-tuples ofpolynomials of the form x+1^{n-1}x+a to their Schur-Szego product, seebelow. As a corollary we obtain that every N nx has all roots real andnon-positive. Additionally, we present an explicit formula for the density andthe distribution function of the asymptotic root-counting measure of thepolynomial sequence {N nx}.



Author: Vladimir Kostov, Boris Shapiro

Source: https://arxiv.org/







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