The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.

We consider the asymptotic behavior of the incomplete gamma functions fl(\Gammaa; \Gammaz) and \Gamma(\Gammaa; \Gammaz) as a !1. Uniform expansions are needed to describe the transition area z a, in which case error functions are used as main approximants. We use integral representations of the incomplete gamma functions and derive a uniform expansion by applying techniques used for the existing uniform expansions for fl(a; z) and \Gamma(a; z). The result is compared with Olver's uniform expansion for the generalized exponential integral. A numerical verification of the expansion is given in a final section.

We consider the eigenvalue equation for the Laplace-Beltrami operator acting on scalar functions on the non-compact Eguchi-Hanson space. The corresponding differential equation is reducible to a confluent Heun equation with Ince symbol [0, 2, 12]. We construct approximations for the eigenfunctions and their asymptotic scattering phases with the help of the Liouville-Green approximation (WKB). Furthermore, for specific discrete eigenvalues obtained by a continued T-fraction we construct the solution by the Frobenius method and determine its scattering phase by a monodromy computation. 1

The paper discusses asymptotic methods for integrals, in particular uniform approximations. We discuss several examples, where we apply the results to Tricomi's \Psi\Gammafunction, in particular we consider an expansion of Tricomi-Carlitz polynomials in terms of Hermite polynomials. We mention other recent expansions for orthogonal polynomials that do not satisfy a differential equation, and for which methods based on integral representations produce powerful uniform asymptotic expansions. 1991 Mathematics Subject Classification: 41A60, 33B20, 33C10, 33C45, 11B73, 30E15. Keywords and Phrases: uniform asymptotic expansions, Tricomi's \Psi\Gammafunction, Kummer functions, confluent hypergeometric functions, Whittaker functions, Hermite polynomials, Tricomi-Carlitz polynomials. Note: Work carried out under project MAS2.8 Exploratory research. Extended version of a paper presented at the Conference Tricomi's Ideas and Contemporary Applied Mathematics to celebrate the 100th anniversary o...