|XXXIV Workshop on Geometric Methods in Physics||28.06-4.07.2015|
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Four types of orthogonal polynomials of affine Weyl groups.
The link between the Fourier calculus of the four families of special functions associated to root systems, $C-$, $S-$, $S^s-$ and $S^l-$functions, and the four families of the induced orthogonal polynomials is discussed. The affine Weyl groups corresponding to the root systems of simple Lie algebras are recalled and sign homomorphisms, which allow general explicit description of the orbit functions, are described. Both continuous and discrete orthogonality of the four types of orbit functions are detailed for each type and the weights, which label the orthogonal functions, are chosen for each type of function and orthogonality separately. The four types of Chebyshev-like orthogonal polynomials, induced by the four types of orbit functions, inherit the discrete and continuous orthogonality from the orbit functions. The discrete and continuous orthogonality of the polynomials are explicitly formulated; their application for the development of the related Fourier methods and possible physical applications are discussed.