Narayana numbers and Schur-Szego composition

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 = 1 n C k−1 n C k n where C i j stands for the usual binomial coefficient, i.e. C j = j! i!(j−i)! . They count Dyck paths of length n and with exactly k peaks, see e.g. [13] and they appeared recently in a number of different combinatorial situations, [5, 11, 16]. Strangely enough Narayana polynomials also occur as limits as n → ∞ of the sequences of eigenpolynomials of the Schur-Szegő composition map sending (n − 1)-tuples of polynomials of the form (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)}.