Two point masses perturbation of regular moment functionals

Abstract

When sigma is a regular moment functional, we consider tau := sigma + lambda(1) delta(x - a(1)) + lambda(2) delta(x - a(2)), where lambda(1),lambda(2) is an element of C, a(1),a(2) is an element of R, and a(1) not equal a(2). We first find a necessary and sufficient condition for tau to be regular (or positive-definite when sigma is positive-definite) and then express orthogonal polynomials {R(n)(x)}(n=0)(infinity) relative to tau in terms of orthogonal polynomials {P-n(x)}(n=0)(infinity) relative to sigma. When both sigma and tau are positive-definite, we investigate the relations between zeros of {P-n(x)}(n=0)(infinity) and {R(n)(x)}(n=0)(infinity). Finally, when sigma is semi-classical, we show that tau 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)(infinity), and the class number of tau.