Higher-Order Three-Term Recurrences and Asymptotics of Multiple Orthogonal Polynomials

Abstract

The asymptotic theory is developed for polynomial sequences that are generated by the three-term higher-order recurrence $$Q_{n+1}=zQ_{n}-a_{n-p+1}Q_{n-p},\quad p\in\mathbb{N},n\geq p,$$ where z is a complex variable and the coefficients ak are positive and satisfy the perturbation condition ∑n=1∞|an−a|<∞. Our results generalize known results for p=1, that is, for orthogonal polynomial sequences on the real line that belong to the Blumenthal–Nevai class. As is known, for p≥2, the role of the interval is replaced by a starlike set S of p+1 rays emanating from the origin on which the Qn satisfy a multiple orthogonality condition involving p measures. Here we obtain strong asymptotics for the Qn in the complex plane outside the common support of these measures as well as on the (finite) open rays of their support. In so doing, we obtain an extension of Weyl’s famous theorem dealing with compact perturbations of bounded self-adjoint operators. Furthermore, we derive generalizations of the classical Szegő functions, and we show that there is an underlying Nikishin system hierarchy for the orthogonality measures that is related to the Weyl functions. Our results also have application to Hermite–Pade approximants as well as to vector continued fractions.