Two point masses perturbation of regular moment functionals
Let \(\sigma\) denote a regular moment functional. Then the authors consider \(\tau:=\sigma+\lambda_1\delta(x-a_1)+\lambda_2\delta(x-a_2)\), where \(\lambda_1,\lambda_2\in\mathbb{C}\), \(a_1,a_2\in\mathbb{R}\), and \(a_1\neq a_2\). A necessary and sufficient condition is given for \(\tau\) to be regular (or positive-definite when \(\sigma\) is positive-definite). Further the orthogonal polynomials \(\{R_n(x)\}_{n=0}^\infty\) with respect to \(\tau\) are given in terms of the orthogonal polynomials \(\{P_n(x)\}_{n=0}^\infty\) relative to \(\sigma\). When both \(\sigma\) and \(\tau\) are positive-definite, the relations between zeros of \(\{P_n(x)\}_{n=0}^\infty\) and \(\{R_n(x)\}_{n=0}^\infty\) are investigated. Finally it is shown that if \(\sigma\) is semi-classical then \(\tau\) is also semi-classical. In that case the structure relation, the second-order differential equation satisfied by the semi-classical orthogonal polynomials \(\{R_n(x)\}_{n=0}^\infty\), and the class number of \(\tau\) are given.
scientific article; zbMATH DE number 1029287
scientific article