Q-operator and Factorised Separation Chain for Jack Polynomials

Applying Baxter’s method of the Q-operator to the set of Sekiguchi’s commuting partial differential operators we show that Jack polynomials P (1/g) λ (x1, . . . , xn) are eigenfunctions of a one-parameter family of integral operators Qz. The operators Qz are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Qzk we construct an integral operator Sn factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator Sn admits a factorisation described in terms of restricted Jack polynomials P (1/g) λ (x1, . . . , xk, 1, . . . , 1). Using the operator Qz for z = 0 we give a simple derivation of a previously known integral representation for Jack polynomials.