Heckman–Opdam's Jacobi polynomials for the BCn root system and generalized spherical functions

JACK POLYNOMIALS FOR THE BCn ROOT SYSTEM AND GENERALIZED SPHERICAL FUNCTIONS

Functions on the double coset space K\G/K, where G is a group and K is a subgroup of G are called spherical functions. One can also study functions on G/K with values in a representation V of G which are equivariant with respect to the left action of K. This more general class of functions may be called vector valued spherical functions. The theory of such functions was developed by Harish-Chan...

Generalized Jacobi polynomials/functions and their applications

We introduce a family of generalized Jacobi polynomials/functions with indexes α,β ∈ R which are mutually orthogonal with respect to the corresponding Jacobi weights and which inherit selected important properties of the classical Jacobi polynomials. We establish their basic approximation properties in suitably weighted Sobolev spaces. As an example of their applications, we show that the gener...

An Integral Formula for Generalized Gegenbauer Polynomials and Jacobi Polynomials

Product Formula for Jacobi Polynomials, Spherical Harmonics and Generalized Bessel Function of Dihedral Type

Abstract. We work out the expression of the generalized Bessel function of B2-type derived in [4]. This is done using Dijskma and Koornwinder’s product formula for Jacobi polynomials and the obtained expression is given by multiple integrals involving only a normalized modified Bessel function and two symmetric Beta distributions. We think of that expression as the major step toward the explici...

Generating Functions of Jacobi Polynomials

Multiplicative renormalization method (MRM) for deriving generating functions of orthogonal polynomials is introduced by Asai–Kubo– Kuo. They and Namli gave complete lists of MRM-applicable measures for MRM-factors h(x) = ex and (1 − x)−κ. In this paper, MRM-factors h(x) for which the beta distribution B(p, q) over [0, 1] is MRM-applicable are determined. In other words, all generating function...