Orthogonal polynomials related to some Jacobi-type pencils

2015/07/28 by Zagorodnyuk, Sergey M.
#42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1508.01794

Abstract

In this paper we study a generalization of the class of orthogonal polynomials on the real line. These polynomials satisfy the following relation: (J₅ - λJ₃) p(λ) = 0, where J₃ is a Jacobi matrix and J₅ is a semi-infinite real symmetric five-diagonal matrix with positive numbers on the second subdiagonal, p(λ) = (p₀(λ), p₁(λ), p₂(λ),⋯)^T, the superscript T means the transposition, with the initial conditions p₀(λ) = 1, p₁(λ) = αλ+ β, α> 0, β∈ℝ. Some orthonormality conditions for the polynomials \ p_n(λ) \_n=0^∞ are obtained. An explicit example of such polynomials is constructed.

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