Orthogonal functions satisfying a second-order differential equation

Let {ϕn}n=0∞ be a sequence of functions satisfying a second-order differential equation of the formαϕn″+βϕn′+(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.