New York Journal of Mathematics
Volume 9 (2003) 117-136


Sergey M. Zagorodnyuk

On generalized Jacobi matrices and orthogonal polynomials

Published: August 19, 2003
Keywords: orthogonal polynomials, difference equation
Subject: 42C05, 33C45

We consider systems of polynomials {pn (λ)}n=0 which satisfy a recurrence relation that can be written in a matrix form:
J p (λ) = λN p (λ) , p = ( p0 (λ), p1 (λ), ... )T,
λ ∈ C, N ∈ N, J is a (2N+1)-diagonal, semi-infinite, 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