The paper discusses asymptotic methods for integrals, in particular uniform approximations. We discuss several examples, where we apply the results to Tricomi's $Psi-$function, 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.

Asymptotic approximations, asymptotic expansions (steepest descent, etc.) (msc 41A60), Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) (msc 33B20), Bessel and Airy functions, cylinder functions, ${}_0F_1$ (msc 33C10), Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) (msc 33C45), Bell and Stirling numbers (msc 11B73), Asymptotic representations in the complex domain (msc 30E15)
CWI
Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Temme, N.M. (1998). Recent problems from uniform asymptotic analysis of integral in particular in connection with Tricomi's $ Psi $ -function. Modelling, Analysis and Simulation [MAS]. CWI.