Applications of a Laplace–Beltrami operator for Jack polynomials

An explicit computation is made for a Laplace–Beltrami type operator for Jack polynomials. As applications we obtain: combinatorial formula, determinantal formula and raising operator formula for Jack polynomials, as well as an iterative formula for the Littlewood–Richardson coefficients. One special case of our results implies Mimachi–Yamada’s result on Jack polynomials of rectangular shapes.