An iterative formula for the Kostka-Foulkes polynomials

An iterative formula for the Kostka-Foulkes polynomials is given using the vertex operator realization of the Hall-Littlewood polynomials. The operational formula can handle large Kostka-Foulkes polynomials, and a stability property for the Kostka-Foulkes polynomials is shown. We also use our algorithm to give a formula of $K_{\lambda \mu }(t)$ for $\mu$ being hook-shaped.