Abstract
<p>The aim of this paper is to make an explicit link between the Weyl-orbit functions and the corresponding polynomials, on the one hand, and to several other families of special functions and orthogonal polynomials on the other. The cornerstone is the connection that is made between the one-variable orbit functions of <em>A<sub>1</sub></em> and the four kinds of Chebyshev polynomials. It is shown that there exists a similar connection for the two-variable orbit functions of <em>A<sub>2</sub></em> and a specific version of two variable Jacobi polynomials. The connection with recently studied <em>G<sub>2</sub></em>-polynomials is established. Formulas for connection between the four types of orbit functions of <em>B<sub>n</sub></em> or <em>C<sub>n</sub></em> and the (anti)symmetric multivariate cosine and sine functions are explicitly derived.</p>
Highlights
Special functions associated with the root systems of simple Lie algebras, e.g. Weyl-orbit functions, play an important role in several domains of mathematics and theoretical physics, in particular in representation theory, harmonic analysis, numerical integration and conformal field theory
The collection of Weyl-orbit functions includes four different families of functions called C, S, Ss– and Sl–functions [6, 15, 16, 25]. They are induced from the sign homomorphisms of the Weyl groups of geometric symmetries related to the underlying Lie algebras
The symmetric C–functions and antisymmetric S– functions appear in the representation theory of simple Lie algebras [32, 34]; the S–functions appear in the Weyl character formula and every character of irreducible representations of simple Lie algebra can be written as a linear combination of C–functions
Summary
Special functions associated with the root systems of simple Lie algebras, e.g. Weyl-orbit functions, play an important role in several domains of mathematics and theoretical physics, in particular in representation theory, harmonic analysis, numerical integration and conformal field theory. The collection of Weyl-orbit functions includes four different families of functions called C–, S–, Ss– and Sl–functions [6, 15, 16, 25] They are induced from the sign homomorphisms of the Weyl groups of geometric symmetries related to the underlying Lie algebras. In [24], it is shown that there are analogous formulas for numerical integration, for multivariate functions, that depend on the Weyl group of the simple Lie algebra An and the corresponding C– and S–functions. It can be shown that the C–, S–, Ss– and Sl–functions arising in connection with simple Lie algebras Bn and Cn become, up to a constant, (anti)symmetric multivariate cosine functions and (anti)symmetric multivariate sine functions [17] Note that these generalizations lead to multivariate analogues of Chebyshev polynomials and are used to derive optimal cubature formulas.
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