  Using vertex operator we study Macdonald symmetric functions of rectangular
shapes and their connection with the q-Dyson Laurent polynomial. We find a
vertex operator realization of Macdonald functions and thus give a generalized
Frobenius formula for them. As byproducts of the realization, we find a q-Dyson
constant term orthogonality relation which generalizes a conjecture due to
Kadell in 2000, and we generalize Matsumoto's hyperdeterminant formula for
rectangular Jack functions to Macdonald functions.
