[Paper Review] A unified construction of generalised classical polynomials associated with operators of Calogero-Sutherland type

Martin Hallnäs, Edwin Langmann|arXiv (Cornell University)|Mar 30, 2007

Algebraic structures and combinatorial models34 references4 citations

TL;DR

This paper presents a unified, explicit construction of generalized classical polynomials—such as Hermite, Laguerre, Jacobi, and Bessel types—as eigenfunctions of Calogero-Sutherland-type operators and their deformations. The key contribution is a family of series representations labeled by pairs of non-negative integers $(M,\tilde{M})$, with the simplest form achieved by minimizing $M + \tilde{M}$, enabling efficient computation and characterization of Jack polynomials via Young diagram coverings.

ABSTRACT

In this paper we consider a large class of many-variable polynomials which contains generalisations of the classical Hermite, Laguerre, Jacobi and Bessel polynomials as special cases, and which occur as the polynomial part in the eigenfunctions of Calogero-Sutherland type operators and their deformations recently found and studied by Chalykh, Feigin, Sergeev, and Veselov. We present a unified and explicit construction of all these polynomials.

Motivation & Objective

• To develop a unified framework for constructing generalized classical polynomials arising as eigenfunctions of Calogero-Sutherland-type operators and their deformations.
• To extend previous constructions in special cases to a full class of symmetric polynomials including Hermite, Laguerre, Jacobi, and Bessel types.
• To provide explicit, finite series representations for these polynomials using a pair of non-negative integers $(M,\tilde{M})$, minimizing the number of terms.

Proposed method

• A generalized eigenfunction construction is applied to Calogero-Sutherland-type operators and their deformations, yielding polynomial eigenfunctions.
• The method employs a triangular expansion in terms of Jack polynomials, with coefficients determined by spectral non-degeneracy conditions.
• The construction is extended to deformed systems using super Jack polynomials as a basis, ensuring completeness when $M \geq N$ and $\tilde{M} \geq \tilde{N}$.
• A labeling scheme via $(M,\tilde{M})$ pairs allows for multiple equivalent series representations of the same polynomial.
• The minimal representation is selected by minimizing $M + \tilde{M}$, reducing computational complexity.
• The approach is validated by showing that Jack polynomials admit such a representation if their Young diagram can be covered by two rectangular diagrams of sizes $M \times m$ and $\tilde{m} \times \tilde{M}$.

Experimental results

Research questions

• RQ1 Can a unified construction be developed for generalized classical polynomials associated with Calogero-Sutherland-type operators across all classical families?
• RQ2 What is the role of the $(M,\tilde{M})$ labeling in minimizing the number of terms in series representations of these polynomials?
• RQ3 How does the covering of a Young diagram by two rectangular blocks relate to the existence of a minimal-term representation for Jack polynomials?
• RQ4 Under what conditions do the reduced eigenfunctions of deformed Calogero-Sutherland operators form a complete basis in the space $\Lambda_{N,\tilde{N},\kappa}$?
• RQ5 How does the spectral non-degeneracy of eigenvalues affect the validity and structure of the constructed polynomial eigenfunctions?

Key findings

• The construction yields a unified method to generate generalized classical polynomials, including Hermite, Laguerre, Jacobi, and Bessel types, as eigenfunctions of Calogero-Sutherland-type operators.
• Each polynomial admits multiple series representations labeled by pairs $(M,\tilde{M})$, with the simplest form obtained by minimizing $M + \tilde{M}$.
• A Jack polynomial has a $(M,\tilde{M})$-representation if its Young diagram can be covered by two rectangular blocks of sizes $M \times m$ and $\tilde{m} \times \tilde{M}$, respectively.
• The reduced eigenfunctions constructed are proportional to standard Jack polynomials, with normalization factor $b_{\boldsymbol{\lambda}}$, ensuring consistency with known orthogonal polynomial theory.
• For deformed systems, the reduced eigenfunctions form a complete basis of $\Lambda_{N,\tilde{N},\kappa}$ if $M \geq N$ and $\tilde{M} \geq \tilde{N}$, and span a proper subspace otherwise.
• The method provides a systematic way to compute these polynomials with minimal term count, significantly improving efficiency for high-degree or complex cases.