arXiv subjects

Max van Horssen

Publications and source records attributed to Max van Horssen.

4 recordsLinked to original sources

arXiv2026

An elementary approach to non-symmetric shift operators and their q-analogs

We give an algebraic construction of shift operators for the non-symmetric Heckman-Opdam polynomials and the non-symmetric Macdonald-Koornwinder polynomials. To each linear character of the finite Weyl group, we associate forward and backward shift operators, which are differential-reflection and difference-reflection operators that satisfy certain transmutation relations with the (Dunkl-)Cherednik operators. In the Heckman-Opdam case, the construction recovers the non-symmetric shift operators of Opdam and Toledano Laredo for the sign character. Furthermore, in rank one, we recover the rank-one non-symmetric shift operators previously obtained by the authors and Schl\"osser.

math.RT↗

arXiv2025

Non-Symmetric Askey--Wilson Shift Operators

We classify the shift operators for the symmetric Askey-Wilson polynomials and construct shift operators for the non-symmetric Askey-Wilson polynomials using two decompositions of non-symmetric Askey-Wilson polynomials in terms of symmetric ones. These shift operators are difference-reflection operators, and we discuss the conditions under which they restrict to shift operators for the symmetric Askey-Wilson polynomials. We use them to compute the norms of the non-symmetric Askey-Wilson polynomials and compute their specialisations for $q\to1$. These turn out to be shift operators for the non-symmetric Heckman-Opdam polynomials of type $BC_1$ that have recently been found.

math.CA↗

arXiv2024

Non-symmetric Jacobi polynomials of type $BC_{1}$ as vector-valued polynomials Part 1: spherical functions

We study non-symmetric Jacobi polynomials of type $BC_{1}$ by means of vector-valued and matrix-valued orthogonal polynomials. The interpretation as matrix-valued orthogonal polynomials yields a new expression of the non-symmetric Jacobi polynomials of type $BC_1$ in terms of the symmetric Jacobi polynomials of type $BC_{1}$. In this interpretation, the Cherednik operator, that has the non-symmetric Jacobi polynomials as eigenfunctions, corresponds to two shift operators for the symmetric Jacobi polynomials of type $BC_{1}$. We show that the non-symmetric Jacobi polynomials of type $BC_{1}$ with so-called geometric root multiplicities, interpreted as vector-valued polynomials, can be identified with spherical functions on the sphere $S^{2m+1}=\mathrm{Spin}(2m+2)/\mathrm{Spin}(2m+1)$ associated with the fundamental spin-representation of $\mathrm{Spin}(2m+1)$. The Cherednik operator corresponds to the Dirac operator for the spinors on $S^{2m+1}$ in this interpretation.

math.CA↗

arXiv2024

Non-symmetric Jacobi polynomials of type $BC_1$ as vector-valued polynomials Part 2: Shift operators

We study non-symmetric Jacobi polynomials of type $BC_1$ by means of vector-valued and matrix-valued orthogonal polynomials. The interpretation as matrix-valued orthogonal polynomials allows us to introduce shift operators for the non-symmetric Jacobi polynomials. The shift operators are differential-reflection operators and we present four of these operators that are fundamental in the sense that they generate all shift operators. Moreover, the symmetrizations of these fundamental shift operators are the fundamental shift operators for the symmetric Jacobi polynomials of type $BC_1$. For the realization of non-symmetric Jacobi polynomials of type $BC_1$ as invariant $\mathbb{C}^2$-valued Laurent polynomials, we introduce a homomorphism that is analogous to the Harish-Chandra homomorphism for the symmetric Jacobi polynomials of type $BC_1$. For geometric root multiplicities, the non-symmetric Jacobi polynomials of type $BC_1$ can be interpreted as spherical functions and we show that our Harish-Chandra homomorphism in this context is related to the Lepowsky homomorphism via the radial part map.

math.CA↗