Spin Kostka polynomials and vertex operators

An algebraic iterative formula for the spin Kostka-Foulkes polynomial $K^-_{\xi\mu}(t)$ is given using vertex operator realizations of Hall-Littlewood symmetric functions and Schur's Q-functions. Based on the operational formula, more favorable properties are obtained parallel to the Kostka polynomial. In particular, we obtain some formulae for the number of (unshifted) marked tableaux. As an application, we confirmed a conjecture of Aokage on the expansion of the Schur $P$-function in terms of Schur functions. Tables of $K^-_{\xi\mu}(t)$ for $|\xi|\leq6$ are listed.