A generalization of the Chebyshev polynomials

In this paper we study polynomials that are orthogonal with respect to a weight function which is zero on a set of positive measure. These were initially introduced by Akhiezer as a generalization of the Chebyshev polynomials where the interval of orthogonality is [-1, alpha] boolean OR [beta,1]. Here, this concept is extended and the interval is the union of g + 1 disjoint intervals, [-1, alpha(1)]boolean ORj=1g-1[beta(j), alpha(j+1)]boolean OR [beta(g), 1], denoted by E. Starting from a suitably chosen weight function p, and the three-term recurrence relation satisfied by the polynomials, a hyperelliptic Riemann surface is defined, from which we construct representations for both the polynomials of the first (P-n) and second kind (Q(n)), respectively, in terms of the Riemann theta function of the surface. Explicit expressions for the recurrence coefficients a(n) and b(n) are found in terms of theta functions. The second-order ordinary differential equation, where P-n and Q(n)/w (where w is the Stieltjes transform of the weight) are linearly independent solutions, is found. The simpler case, where g = 1, is extensively dealt with and the reduction to the Chebyshev polynomials in the limiting situation, alpha --> beta, where the two intervals merge into one, is demonstrated. We also show that p(x)k(n)(x, x)/n for x epsilon E, where k(n) (X, X) is the reproducing kernel at coincidence, tends to the equilibrium density of the set E, as n --> infinity.