CMA Research Report

MRR03-004

Q-operator and factorised separation chain for Jack's symmetric polynomials

Abstract:  Applying Baxter's method of the Q-operator to the set of Sekiguchi's commuting partial differential operators we show that Jack's symmetric polynomials P_λ^(1/g)(x₁,...,x_n) are eigenfunctions of a one-parameter family of integral operators Q_z. The operators Q_z are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Q_zk we construct an integral operator S_n factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator S_n admits a factorisation described in terms of restricted Jack polynomials P_λ^(1/g)(x₁,...,x_k,1,...,1). Using the operator Q_z for z = 0 we give a simple derivation of a previously known integral representation for Jack polynomials.

AMS Classification:  70H06, 33, 37J35
Date:  17 June 2003

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