Q-operator and factorised separation chain for Jack polynomials

Original · EN

Applying Baxter's method of the Q-operator to the set of Sekiguchi's commuting partial differential operators we show that Jack polynomials P(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(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.