 

Sergey M. Zagorodnyuk
On generalized Jacobi matrices and orthogonal polynomials


Published: 
August 19, 2003

Keywords: 
orthogonal polynomials, difference equation 
Subject: 
42C05, 33C45 


Abstract
We consider systems of polynomials {p_{n} (λ)}_{n=0}^{∞} which
satisfy a recurrence relation that can be written in a matrix form:
J p
(λ) = λ^{N} p (λ) , p = ( p_{0} (λ), p_{1} (λ),
... )^{T},
λ ∈ C, N ∈ N, J is a (2N+1)diagonal,
semiinfinite, Hermitian complex numerical matrix. For such systems we
obtained orthonormality relations on radial rays. To prove these
relations we used the standard method of scattering theory. We showed
that these relations are characteristic. From the relations it is easily
shown that systems of orthonormal polynomials on the real line, systems of
Sobolev orthogonal polynomials with discrete measure at zero, systems or
orthonormal polynomials on radial rays with a scalar measure are such
systems of polynomials. Also we consider a connection with matrix
orthonormal polynomials on the real line.


Author information
Department of Mathematics and Informatics, School of Mathematics and Mechanics, Karazin Kharkiv National University, Kharkiv 61077, Ukraine
Sergey.M.Zagorodnyuk@univer.kharkov.ua

