Quasi-definiteness of generalized Uvarov transforms of moment functionals

Abstract

When $\sigma$ is a quasi-definite moment functional with the monic orthogonal polynomial system $\{P_{n}(x)\}_{n=0}^{\infty}$, we consider a point masses perturbation $\tau$ of $\sigma$ given by $\tau :=\sigma +\lambda \sum_{l=1}^{m}\sum_{k=0}^{m_{l}}({(-1)^{k}u_{lk}}/{k!})\delta^{(k)}(x-c_{l})$, where $\lambda,u_{lk}$, and $c_l$ are constants with $c_i\neq c_j$ for $i\neq j$. That is, $\tau$ is a generalized Uvarov transform of $\sigma$ satisfying $A(x)\tau = A(x)\sigma$, where $A(x) =\prod_{l=1}^{m}(x-c_{l})^{m_{l}+1}$. We find necessary and sufficient conditions for $\tau$ to be quasi-definite. We also discuss various properties of monic orthogonal polynomial system $\{R_{n}(x)\}_{n=0}^{\infty}$ relative to $\tau$ including two examples.