Narayana numbers and Schur-Szego composition

TL;DR

This paper offers a new interpretation of Narayana polynomials related to Dyck paths and reveals their connection to eigenpolynomials of the Schur-Szego composition map, establishing their real, non-positive roots and describing their asymptotic root distribution.

Contribution

It introduces a novel interpretation of Narayana polynomials and links them to eigenpolynomials of the Schur-Szego composition, providing new insights into their roots and asymptotic behavior.

Findings

Narayana polynomials have all roots real and non-positive.

Established the connection between Narayana polynomials and eigenpolynomials of the Schur-Szego map.

Derived explicit formulas for the density and distribution of the asymptotic root measure.

Abstract

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,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 N_n(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 {N_n(x)}.