  An iterative formula for the Green polynomial is given using the vertex
operator realization of the Hall-Littlewood functions. Based on this, (1) a
general combinatorial formula of the Green polynomial is given; (2) several
compact formulas are given for Green's polynomials associated with upper
partitions of length $\leq 3$ and the diagonal lengths $\leq 3$; (3) a
Murnaghan-Nakayama type formula for the Green polynomial is obtained; and (4)
an iterative formula is derived for the bitrace of the finite general linear
group $G$ and the Iwahori-Hecke algebra of type $A$ on the permutation module
of $G$ by its Borel subgroup.
