Two point masses perturbation of regular moment functionals

Abstract

When σ is a regular moment functional, we consider τ : = σ + λ 1 δ( x − a 1) + λ 2 δ( x − a 2), where λ 1, λ 2 ϵ ¢C, a 1, a 2 ϵ ¢R , and a 1 ≠ a 2. We first find a necessary and sufficient condition for τ to be regular (or positive-definite when σ is positive-definite) and then express orthogonal polynomials { R n ( x)} n = 0 ∞ relative to τ in terms of orthogonal polynomials { P n ( x)} n = 0 ∞ relative to σ. When both σ and τ are positive-definite, we investigate the relations between zeros of { P n ( x)} n = 0 ∞ and { R n ( x)} n = 0 ∞. Finally, when σ is semi-classical, we show that τ is also semi-classical and give the structure relation, second-order differential equation satisfied by the semi-classical orthogonal polynomials { R n ( x)} n = 0 ∞, and the class number of τ.