Kostka polynomials, which have a close connection with Green polynomials of finite general linear groups $GL_n(F_q)$, are defined as the transition matrix between Schur functions and Hall-Littlewood functions. The notion of Green polynomials are generalized, by a geometric method, to other classical groups, such as $Sp_{2n}(F_q)$ or $SO_{2n+1}(F_q)$. They are characterized by the corresponding Weyl groups and some additiona data. The complex reflection group $G(e,1,n)$ is a generalization of the symmetric group $S_n$, and of the Weyl group of type $B_n$. We construct Schur functions and Hall-Littlewood functions associated to $G(e,1,n)$ generalizing the case of type $A$, and define Kostka functions as a transition matrix between them. As a special case, this gives a combinatorial approach for Green polynomials of type $B_n$ or $C_n$.