Publications

We obtain uniform asymptotic approximations for the monic Meixner–Sobolev polynomials Sn(x). These approximations for n → ∞, are uniformly valid for x/n restricted to certain intervals, and are in terms of Airy functions. We also give asymptotic approximations for the location of the zeros of Sn(x), especially the small and the large zeros are discussed. As a limit case, we also give a new asymptotic approximation for the large zeros of the classical Meixner polynomials.

The method is based on an integral representation in which a hypergeometric function appears in the integrand. After a transformation, the hypergeometric functions can be uniformly approximated by unity, and all that remains are simple integrals for which standard asymptotic methods are used. As far as we are aware, this is the first time that standard uniform asymptotic methods have been used for the Sobolev-class of orthogonal polynomials.

We obtain new uniform asymptotic approximations for integrals with a relatively exponentially small remainder. We illustrate how these results can be used to obtain remainder estimates in the Bleistein method. The method is created to deal with new types of integrals in which the usual methods for remainder estimates fail. As an application, we obtain an asymptotic expansion for

as in |ph λ|≤π/2 uniformly for large |z|.

In this paper, we obtain asymptotic expansions for the Gauss hypergeometric function

, where ej = 0, ± 1, j = 1, 2, 3, as |λ| → ∞. We complete the results of three previous publications [Uniform asymptotic expansions for hypergeometric functions with large parameters I, Anal. Appl. (Singap.) 1 (2003) 111–120; Uniform asymptotic expansions for hypergeometric functions with large parameters II, Anal. Appl. (Singap.) 1 (2003) 121–128; Uniform asymptotic expansions for hypergeometric functions with large parameters III, Anal. Appl. (Singap.) 8 (2010) 199–210], discuss all cases and, what is new, we consider now all critical values of z. For one case, the full details of the well-known Bleistein method are given, since a new technical detail is observed.

  • Computation of the coefficients appearing in the uniform asymptotic expansions of integrals. Farid Khwaja, S: Olde Daalhuis, A. B. Accepted for publication at Studies in Applied Mathematics ( March 2017)

The coefficients that appear in uniform asymptotic expansions for integrals are typically very complicated. In the existing literature the majority of the work only give the first two coefficients. In a limited number of papers where more coefficients are given the evaluation of the coefficients near the coalescence points is normally highly numerically unstable. In this paper, we illustrate how well-known Cauchy type integral representations can be used to compute the coefficients in a very stable and efficient manner. We discuss the cases: (i) two coalescing saddles, (ii) two saddles coalesce with two branch points, (iii) a saddle point near an endpoint of the interval of integration.

As a special case of (ii) we give a new uniform asymptotic expansion for Jacobi polynomials P(α,β)n(z) in terms of Laguerre polynomials L(α)n(x) as n→∞ that holds uniformly for z near 1. Several numerical illustrations are included.