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,α]∪[β,1]. Here, this concept is extended and the interval is the union of g + 1 disjoint intervals, [-1,α₁]∪_{j = 1}^g-1[β_j,α_{j + 1}]∪[β_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, α→β, where the two intervals merge into one, is demonstrated. We also show that p(x)k_n(x,x)/n for x∊E, where k_n(x,x) is the reproducing kernel at coincidence, tends to the equilibrium density of the set E, as n→∞.