Orthogonal polynomials related to some Jacobi-type pencils - Zagorodnyuk, S. M. (Ukrains’kyi Matematychnyi Zhurnal, 2016)
We study a generalization of the class of orthonormal polynomials on the real axis. These polynomials satisfy the followingrelation: $(J_5 λ J_3)\vec{p}(λ) = 0$, where $J_3$ is a Jacobi matrix and $J_5$ is a semi-infinite real symmetric five-diagonal matrix with positive numbers on the second subdiagonal, $\vec{p}(λ) = (p_0(λ ), p_1(λ ), p_2(λ ),...)^T$, the superscript $T$ denotes the operation of transposition with the initial conditions $p_0(λ ) = 1,\; p_1(λ) = α λ + β,\; α > 0, β \in R$. Certain orthonormalityconditions for the polynomials $\{ pn(λ )\}^{∞}_n = 0$ are obtained. An explicit example of these polynomials is constructed.