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Q-operator and factorised separation chain for Jack polynomials

Vadim B. Kuznetsov·Vladimir V. Mangazeev·Evgeny K. Sklyanin
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Abstract

Applying Baxter's method of the Q-operator to the set of Sekiguchi's commuting partial differential operators we show that Jack polynomials P(x_1,...,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_{z_k} 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(x_1,...,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.

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BibTeXRIS

Vadim B. Kuznetsov, Vladimir V. Mangazeev, Evgeny K. Sklyanin. 2003-06-16. Q-operator and factorised separation chain for Jack polynomials. https://doi.org/10.1016/s0019-3577(03)90057-7

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