Difference equations and symmetric polynomials defined by their zeros

Abstract

In this paper, we introduce a new family of symmetric polynomials which depends on a parameter r. They are defined by specifying certain of their zeros. For the parameter values 1/2, 1, and 2 they have an interpretation in terms of Capelli identities. First, we give explicit formulas in some special cases. Then we show that the polynomials can also be defined in terms of difference equations. As a corollary we obtain that their top homogeneous part is a Jack polynomial. This is used to give a new proof of the Pieri formula for Jack polynomials.