Q-operator and factorised separation chain for Jack polynomials |
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Authors: | Vadim B. Kuznetsov Vladimir V. Mangazeev Evgeny K. Sklyanin |
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Affiliation: | a Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK;b Centre for Mathematics and Its Applications, Mathematical Science Institute, Australian National University, Canberra, ACT 0200, Australia;c Department of Mathematics, University of York, York Y010 SDD, UK |
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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 Pλ(1/g) (χ1, …, χn) …, χn) 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 Pλ(1/g) (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. |
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Keywords: | Author Keywords: Jack polynomials integral operators |
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