Quasi-definiteness of generalized Uvarov transforms of moment functionals

Abstract

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