A unified construction of generalised classical polynomials associated with operators of Calogero-Sutherland type
Martin Hallnäs, Edwin Langmann
TL;DR
The paper presents a unified explicit construction of multi-variable polynomials that form eigenfunctions of Calogero-Sutherland type operators and their deformations, thereby unifying generalizations of Hermite, Laguerre, Jacobi and Bessel polynomials. The approach hinges on a ground-state factorization and a triangular action on a basis of functions $f_{oldsymbol{n}}$, enabling reduced eigenfunctions $P_{oldsymbol{n}}$ to be written as convergent series $P_{oldsymbol{n}}=f_{oldsymbol{n}}+ ext{lower terms}$. It systematically extends to deformed CS operators $H_{N, ilde{N}}$, producing families of eigenfunctions labeled by pairs $(M, ilde{M})$ and connecting to Jack polynomials and super Jack polynomials through dualities and fat-hook partitions. The results yield explicit representations, completeness results, and a framework to minimize series complexity by appropriate choices of $(M, ilde{M})$, with broad connections to quasi-exact solvability and generalized orthogonal polynomials. This work advances the exact solution program for quantum many-body systems and enriches the theory of generalized multivariate orthogonal polynomials with concrete constructions and dualities.
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.