# Pieri-type formulas for the non-symmetric Jack polynomials

### P. J. Forrester

University of Melbourne, Parkville, Australia### D. S. McAnally

University of Melbourne, Parkville, Australia

## 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.

## Cite this article

P. J. Forrester, D. S. McAnally, Pieri-type formulas for the non-symmetric Jack polynomials. Comment. Math. Helv. 79 (2004), no. 1, pp. 1–24

DOI 10.1007/S00014-003-0789-2