  We consider in this paper the Hahn polynomials and their application in
numerical methods. The Hahn polynomials are classical discrete orthogonal
polynomials. We analyse the behaviour of these polynomials in the context of
spectral approximation of partial differential equations. We study series
expansions $u=\sum_{n=0}^\infty \hat{u}_n \phi_n$, where the $\phi_n$ are the
Hahn polynomials. We examine the Hahn coefficients and proof spectral accuracy
in some sense. We substantiate our results by numericals tests. Furthermore we
discuss a problem which arise by using the Hahn polynomials in the
approximation of a function $u$, which is linked to the Runge phenomenon. We
suggest two approaches to avoid this problem. These will also be the motivation
and the outlook of further research in the application of discrete orthogonal
polynomials in a spectral method for the numerical solution of hyperbolic
conservation laws.
