Constructive Approximation

, Volume 38, Issue 2, pp 277–309 | Cite as

A Hypergeometric Function Transform and Matrix-Valued Orthogonal Polynomials

  • Wolter Groenevelt
  • Erik Koelink


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.


Hypergeometric function transform Matrix-valued orthogonal polynomials Second-order differential operator Five-term operator Spectral analysis 

Mathematics Subject Classification

33C45 47B25 47E05 


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.DIAMTechnische Universiteit DelftDelftThe Netherlands
  2. 2.IMAPP, FNWIRadboud Universiteit NijmegenNijmegenThe Netherlands

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