Some continuous analogs of expansion in Jacobi polynomials and vector valued hypergeometric orthogonal bases

Details Some continuous analogs of expansion in Jacobi polynomials and vector valued hypergeometric orthogonal bases Some continuous analogs of expansion in Jacobi polynomials and vector valued hypergeometric orthogonal bases

2003-09-27

arxiv.org/abs/math/0309445v2

Mathematics

Classical Analysis and ODEs

27 pages; Addendum 'Perturbations of classical hypergeometric orthogonal systems' is added

Scientific paper

We write spectral decomposition of the hypergeometric differential operator on the contour $Re z=1/2$ (multiplicity of spectrum is 2). As a result, we obtain an integral transform that differs from the Jacobi (or Olevsky) transform. We also try to do a step towards vector-valued special functions and construct a 3F2 orthogonal basis in the space of functions having values in 2-dimensional space. This basis is lying in an analytic continuation of continuous dual Hahn polynomials with respect to number $n$ of a polynomial. In Addendum, we discuss a vector-value analog of the Meixner--Pollachek orthogonal system and also perturbations of Laguerre, Meixner, and Jacobi polynomials.

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