Determinantal Expression and Recursion for Jack Polynomials

We give matrices of which their determinants are the Jack polynomials expanded in terms of the monomial basis. The top row of this matrix is a list of monomial functions, the entries of the sub-diagonal are of the form −(rα + s), with r and s ∈ +, the entries above the sub-diagonal are nonnegative integers, and below all entries are 0. The quasi-triangular nature of this matrix gives a recursion for the Jack polynomials allowing for efficient computation. A specialization of these results yields a determinantal formula for the Schur functions and a recursion for the Kostka numbers.