On Polynomials Related to Powers of the Generating Function of Catalan's Numbers

Catalan's sequence of numbers {C„}% = {1,1,2,5,14,42,...} (nr.1459 and ,4000108 of [14]) emerges in the solution of many combinatorial problems (see [2], [4], [5], and [16] for further references). The moments ju2k of the normalized weight function of Chebyshev's polynomials of the second kind are given by Ck /2 (see, e.g., [3], Lemma 4.3, p. 160 for /= 0, and [17], p. II3). This sequence also shows up in the asymptotic moments of zeros of scaled Laguerre and Hermite polynomials (see [9], eqs. (3.34) and (3.35)). The generating function c(x) Z^=0 Cnx is the solution of the quadratic equation xc(x) c(x) + 1 = 0 with c(0) = 1. Therefore, every positive integer power of c(x) can be written as