A finite-sum representation for solutions for the Jacobi operator
Hugo M. Campos, Vladislav V. Kravchenko
TL;DR
This paper derives a finite-sum formula for solutions of the Jacobi difference equation, enabling better analysis of solution properties and boundedness, with practical examples illustrating its application.
Contribution
It introduces a novel finite-sum representation for Jacobi difference equation solutions based on a nonvanishing particular solution.
Findings
Finite-sum representation for solutions derived
Applications to boundedness of solutions demonstrated
Illustrating examples provided
Abstract
We obtain a finite-sum representation for the general solution of the Jacobi second-order difference equation D(p(n-1)Du(n-1))+q(n)u(n)=l r(n)u(n) in terms of a nonvanishing solution corresponding to some fixed value of the spectral parameter l=l_0. Applications of this representation to some results on the boundedness of solutions are given as well as illustrating examples.
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