A Hypergeometric Function Transform and Matrix-Valued Orthogonal Polynomials
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, leading to orthogonality relations for 2×2-matrix-valued polynomials. These matrix-valued polynomials can be considered as matrix-valued generalizations of Wilson polynomials.
KeywordsHypergeometric function transform Matrix-valued orthogonal polynomials Second-order differential operator Five-term operator Spectral analysis
Mathematics Subject Classification33C45 47B25 47E05
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