Pieri - type formulas for the non - symmetric Jack polynomials

In the theory of symmetric Jack polynomials the coefficients in the expansion of the pth elementary symmetric function ep(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 Eη(z) are explored. Necessary conditions for non-zero coefficients in the expansion of ep(z)Eη(z) as a series in non-symmetric Jack polynomials are given. A known expansion formula for ziEη(z) is rederived by an induction procedure, and this expansion is used to deduce the corresponding result for the expansion of N j=1, j =i zj Eη(z), and consequently the expansion of eN−1(z)Eη(z). In the general p case the coefficients for special terms in the expansion are presented.