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 following relation: (J_5 λ J_3)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, 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 R. Certain orthonormality conditions for the polynomials { pn(λ )}^∞_n = 0 are obtained. An explicit example of these polynomials is constructed.