Difference Equations and Symmetric Polynomials Defined by Their Zeros
نویسندگان
چکیده
In this paper, we are starting a systematic analysis of a class of symmetric polynomials which, in full generality,was introduced in [Sa]. The main features of these functions are that they are defined by vanishing conditions and that they are nonhomogeneous. They depend on several parameters, but we are studying mainly a certain subfamily which is indexed by one parameter, r. As a special case, we obtain for r = 1 the factorial Schur functions discovered by Biedenharn and Louck [BL]. Our main result is that for general r these functions are eigenvalues of difference operators,which are difference analogues of the Sekiguchi-Debiard differential operators. Thus the functions under investigation are nonhomogeneous variants of Jack polynomials. More precisely, consider the set of partitions of length n, i.e., sequences of integers (λi) with λ1 ≥ · · · ≥ λn ≥ 0. The weight |λ| of a partition λ is the sum of its parts λi. Choose a vector ρ ∈ C which has to satisfy a mild condition. Then, for every λ, there is (up to a constant) a unique symmetric polynomial Pλ of degree at most d which satisfies the following vanishing condition:
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