Several orthogonal polynomials have limit forms in which Hermite polynomials show up. Examples are limits with respect to certain parameters of the Jacobi and Laguerre polynomials. In this paper we are interested in more details of these limits and we give asymptotic representations of several orthogonal polynomials in terms of Hermite polynomials. In fact we give finite exact representations that have an asymptotic character. From these representations the well-known limits can be derived easily. Approximations of the zeros of the Gegenbauer polynomials $C_n^{gamma(x)$ and Laguerre polynomials $L_n^{alpha(x)$ are derived (for large values of $gamma$ and $alpha$, respectively) in terms of zeros of the Hermite polynomials and compared with numerical values. We also consider the Jacobi polynomials and the so-called Tricomi-Carlitz polynomials.

Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) (msc 33C45), Asymptotic approximations, asymptotic expansions (steepest descent, etc.) (msc 41A60), Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) (msc 30C15), Approximation by polynomials (msc 41A10)
CWI
Modelling, Analysis and Simulation [MAS]
Computational Dynamics

López, J.L, & Temme, N.M. (1999). Approximations of orthogonal polynomials in terms of Hermite polynomials. Modelling, Analysis and Simulation [MAS]. CWI.