Study of the asymptotics of orthogonal polynomials leads naturally to a division of the plane into three regions: the complement of the closed interval I that contains the zeros (perhaps after rescaling), the interior of the interval I, and the endpoints of I. For the classical polynomials, integral representations or differential equations can be used, as in Chapter 13. However, the discrete orthogonal polynomials do not satisfy a differential equation, and the Chebyshev–Hahn polynomials have no one-dimensional integral representation.

Each of the two methods illustrated in this chapter begins from the easiest part of the problem, a determination of the leading asymptotics in the exterior region, i.e. the complement of the interval. In the first section, we compute such an approximation in the case of Hermite polynomials. Corresponding results for Laguerre and Jacobi polynomials are stated, with the details left to the exercises.

The recently introduced Riemann–Hilbert method is a powerful tool for obtaining global asymptotics, i.e. asymptotic approximations in each of a few regions that cover the complex plane, for the orthogonal polynomials associated with certain types of weight function.

On the other hand, there are many orthogonal polynomials for which the weight function is not known or is not unique. In this case, the main tool has to be an associated three-term recurrence relation. We give an indication in the case of Jacobi polynomials. The three-term recurrence relation leads to a general form of the asymptotics on a complex neighborhood of the interval (−1, 1). This form contains several functions that can be determined explicitly by matching to an approximation that is valid on the complement of the closed interval. Corresponding results are stated for Laguerre and Hermite polynomials.

In broad outline, the Riemann–Hilbert method consists in embedding a given object, such as an orthogonal polynomial, as part of the solution of a Riemann–Hilbert problem; the problem of determining a matrix-valued function of a complex variable that is holomorphic off a certain contour Σ and satisfies specified jump conditions across Σ as well as a normalization condition at infinity. Transforming such a problem into a more tractable form can lead to an explicit asymptotic expansion. We illustrate this in the case of Hermite polynomials. The ideas of Section 4.2 are used to set up the Riemann–Hilbert formulation in general.