Determinantal expression and recursion for Jack polynomials (Q1964487)

The Jack polynomials \(J_{\lambda}[x_1,\ldots,x_N;\alpha]\) form a basis for the space of symmetric polynomials in \(N\) variables; the specialization \(\alpha=1\) in \(J_{\lambda}[x;\alpha]\) gives the Schur function \(s_{\lambda}[x]\). The main purpose of the paper under review is to give a matrix with determinant equal to \(J_{\lambda}[x;\alpha]\). The matrix under consideration has only one nonzero subdiagonal. This gives rise to a simple recursion which may be used for the rapid computation of Jack polynomials. As a consequence of these results the authors obtain a new determinantal form for the Schur functions and a recursion for the Kostka numbers which express the coefficients of the Schur functions in terms of the monomial basis. The authors illuminate their results with successfully selected examples.