Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 050, 14 pages      arXiv:1512.03898      https://doi.org/10.3842/SIGMA.2016.050
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

Automorphisms of Algebras and Bochner's Property for Vector Orthogonal Polynomials

Emil Horozov ab
a) Department of Mathematics and Informatics, Sofia University, 5 J. Bourchier Blvd., Sofia 1126, Bulgaria
b) Institute of Mathematics and Informatics, Bulg. Acad. of Sci., Acad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria

Received January 26, 2016, in final form May 12, 2016; Published online May 19, 2016

Abstract
We construct new families of vector orthogonal polynomials that have the property to be eigenfunctions of some differential operator. They are extensions of the Hermite and Laguerre polynomial systems. A third family, whose first member has been found by Y. Ben Cheikh and K. Douak is also constructed. The ideas behind our approach lie in the studies of bispectral operators. We exploit automorphisms of associative algebras which transform elementary vector orthogonal polynomial systems which are eigenfunctions of a differential operator into other systems of this type.

Key words: vector orthogonal polynomials; finite recurrence relations; bispectral problem; Bochner theorem.

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