Pieri-type formulas for the non-symmetric Jack polynomials

Abstract

In the theory of symmetric Jack polynomials the coefficients in the expansion of the $p$th elementary symmetric function $e_p(z)$ times a Jack polynomial expressed as a series in Jack polynomials are known explicitly. Here analogues of this result for the non-symmetric Jack polynomials $E_{\eta }(z)$ are explored. Necessary conditions for non-zero coefficients in the expansion of $e_p(z)E_{\eta }(z)$ as a series in non-symmetric Jack polynomials are given. A known expansion formula for $z_i{E}_{\eta }(z)$ is rederived by an induction procedure, and this expansion is used to deduce the corresponding result for the expansion of ${\prod }_{j=1,j\ne i}^N{z}_j{E}_{\eta }(z)$, and consequently the expansion of $e_{N-1}(z)E_{\eta }(z)$. In the general $p$ case the coefficients for special terms in the expansion are presented.