Narayana numbers and Schur-Szego composition
Abstract
In the present paper we find a new interpretation of Narayana polynomials Nn(x) which are the generating polynomials for the Narayana numbers Nn,k counting Dyck paths of length n and with exactly k peaks. Strangely enough Narayana polynomials also occur as limits as n->oo 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. As a corollary we obtain that every Nn(x) has all roots real and non-positive. Additionally, we present an explicit formula for the density and the distribution function of the asymptotic root-counting measure of the polynomial sequence Nn(x).
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