  We use a new method to study the Laplace-Beltrami type operator on the Fock
space of symmetric functions, and as an example of our explicit computation we
show that the Jack symmetric functions are the only family of eigenvectors of
the differential operator. As applications of this explicit method we find a
combinatorial formula for Jack symmetric functions and the
Littlewood-Richardson coefficients in the Jack case. As further applications,
we obtain a new determinantal formula for Jack symmetric functions. We also
obtained a generalized raising operator formula for Jack symmetric functions,
and a formula for the explicit action of Virasoro operators. Special cases of
our formulas imply Mimachi-Yamada's result on Jack symmetric functions of
rectangular shapes, as well as the explicit formula for Jack functions of two
rows or two columns.
