Three systems of orthogonal polynomials and L2-boundedness of two associated operators

In this paper, we describe three systems of orthogonal polynomials belonging to the class of Meixner-Pollaczek polynomials, and establish some useful connections between them in terms of three basic operators that are related to them. Furthermore, we investigate boundedness properties of two other operators, both as convolution operators in the translation invariant case where we use Fourier transforms and for the weights related to the relevant orthogonal polynomials. We consider only the most important but also simplest case of L2-spaces. However, in subsequent papers, we intend to extend the study to Lp-spaces (1