ELLIPTIC ORTHOGONAL AND EXTREMAL POLYNOMIALS

In this paper we give, in terms of elliptic functions, an explicit rep resentation of the polynomials which are orthogonal on E = [-1, alpha] boolean OR[beta, 1] with respect to a weight function of the form root {(1 - x(2))(x - alpha)(x - beta)}/\rho(x)\ (in fact a more general cla ss of weight functions, not necessarily positive on E, is considered), where rho is an arbitrary polynomial which has no zero in int(E). Als o closed expressions are provided for the recurrence coefficients of t he orthogonal polynomials. Furthermore it is shown that under certain special conditions these orthogonal polynomials are also minimal polyn omials on E with respect to the maximum norm with weight 1/root\rho(x) \. Then, for a large class of weight functions (including the above-me ntioned ones), the support of which consists of two intervals which ha ve non-rational harmonic measure at infinity, it is shown that the rec urrence coefficients of the orthogonal polynomials have an infinite se t of limit points. This fact is rather surprising, because in the sing le-interval case it is well known that the recurrence coefficients of the polynomials orthogonal with respect to a weight function positive almost everywhere on the interval converge. Finally we give the explic it solution of periodic and non-periodic Toda lattices the band struct ure of which consists of two intervals.