The search for differential equations for certain sets of orthogonal polynomials
SPOA VII Proceedings of the seventh Spanish symposium on Orthogonal polynomials and applications
Differential equations having orthogonal polynomial solutions
Journal of Computational and Applied Mathematics
Ordinary Differential Equations
Ordinary Differential Equations
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Let {φn}n=0∞ be a sequence of functions satisfying a second-order differential equation of the form αφ''n+ βφ'n + (σ + λnτ)φn = fn, where α, β, σ, τ, and fn are smooth functions on the real line R, and λn is the eigenvalue parameter. Then we find a necessary and sufficient condition in order for {φn}n=0∞ to be orthogonal relative to a distribution w and then we give a method to find the distributional orthogonalizing weight w. For such an orthogonal function system, we also give a necessary and sufficient condition in order that the derived set {(pφn)'}n=0∞ is orthogonal, which is a generalization of Lewis and Hahn. We also give various examples.