# 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 $E_{η}(z)$ are explored. Necessary conditions for non-zero coefficients in the expansion of $e_{p}(z)E_{η}(z)$ as a series in non-symmetric Jack polynomials are given. A known expansion formula for $z_{i}E_{η}(z)$ is rederived by an induction procedure, and this expansion is used to deduce the corresponding result for the expansion of $∏_{j=1,j=i}z_{j}E_{η}(z)$, and consequently the expansion of $e_{N−1}(z)E_{η}(z)$. 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