Narayana numbers and Schur-Szego{double acute} composition

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} = frac(1, n) C_n^{k - 1} C_n^k where C_jⁱ stands for the usual binomial coefficient, i.e. C_jⁱ = frac(j !, i ! (j - i) !). They count Dyck paths of length n and with exactly k 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 → ∞ of the sequences of eigenpolynomials of the Schur-Szego{double acute} composition map sending (n - 1)-tuples of polynomials of the form (x + 1)^{n - 1} (x + a) to their Schur-Szego{double acute} 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 {N_n (x)}.