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Mathematics > Classical Analysis and ODEs

arXiv:math/0306242 (math)
[Submitted on 16 Jun 2003 (v1), last revised 1 Jul 2003 (this version, v2)]

Title:Q-operator and factorised separation chain for Jack polynomials

Authors: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.
Comments: 26 pages
Subjects: Classical Analysis and ODEs (math.CA); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Exactly Solvable and Integrable Systems (nlin.SI)
MSC classes: 58F07 (Primary) 33 (Secondary)
Cite as: arXiv:math/0306242 [math.CA]
  (or arXiv:math/0306242v2 [math.CA] for this version)
  https://doi.org/10.48550/arXiv.math/0306242
Journal reference: Indag .Math. 14 (2003) 451-482
Related DOI: https://doi.org/10.1016/S0019-3577%2803%2990057-7

Submission history

From: Vadim B. Kuznetsov [view email]
[v1] Mon, 16 Jun 2003 18:51:49 UTC (29 KB)
[v2] Tue, 1 Jul 2003 11:59:09 UTC (29 KB)
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