Quasi-definiteness of Generalized Uvarov Transforms of Moment Functionals

When σ is a quasi-definite moment functional with the monic orthogonal polynomial system {Pn(x)}∞n=0, we consider a point masses perturbation τ of σ given by τ := σ+ λ ∑m l=1 ∑ml k=0((−1) ulk/k!)δ (x− cl), where λ, ulk, and cl are constants with ci = cj for i = j. That is, τ is a generalized Uvarov transform of σ satisfying A(x)τ = A(x)σ, where A(x) = ∏m l=1(x− cl) l. We find necessary and sufficient conditions for τ to be quasi-definite. We also discuss various properties of monic orthogonal polynomial system {Rn(x)}∞n=0 relative to τ including two examples.