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arXiv · 1210.3958

A hypergeometric function transform and matrix-valued orthogonal polynomials

Wolter Groenevelt·Erik Koelink
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Abstract

The spectral decomposition for an explicit second-order differential operator $T$ is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with multiplicity one. The spectral analysis gives rise to a generalized Fourier transform with an explicit hypergeometric function as a kernel. Using Jacobi polynomials the operator $T$ can also be realized as a five-diagonal operator, hence leading to orthogonality relations for $2\times 2$-matrix-valued polynomials. These matrix-valued polynomials can be considered as matrix-valued generalizations of Wilson polynomials.

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Wolter Groenevelt, Erik Koelink. 2012-10-15. A hypergeometric function transform and matrix-valued orthogonal polynomials. https://doi.org/10.1007/s00365-013-9207-1

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