2024/12/06 by Max van Horssen, van Horssen, Max, Maarten van Pruijssen +1
Mathematics · Computer Science · #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2412.05417
We study non-symmetric Jacobi polynomials of type BC1 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 BC1. For the realization of non-symmetric Jacobi polynomials of type BC1 as invariant ℂ2-valued Laurent polynomials, we introduce a homomorphism that is analogous to the Harish-Chandra homomorphism for the symmetric Jacobi polynomials of type BC1. For geometric root multiplicities, the non-symmetric Jacobi polynomials of type BC1 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.