Hermite polynomials are considered as approximants in asymptotic representations of certain other polynomials. Examples are given for polynomials from the Askey scheme of hypergeometric orthogonal polynomials. We also mention that Hermite polynomials can be used as main approximants in uniform asymptotic representations of certain types of integrals and differential equations.

Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) (msc 33C45), Asymptotic approximations, asymptotic expansions (steepest descent, etc.) (msc 41A60), Approximation by polynomials (msc 41A10)
Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Temme, N.M, & López, J.L. (1999). The role of Hermite polynomials in asymptotic analysis. Modelling, Analysis and Simulation [MAS]. CWI.