GENERATING FUNCTIONS OF ORTHOGONAL POLYNOMIALS AND
Abstract: In this paper, we present a more direct way to compute the SzegŲ-Jacobi parameters
from a generating function than that in  and . Our study is motivated by the notions of
one-mode interacting Fock spaces defined in  and Segal-Bargmann transform associated
with non-Gaussian probability measures introduced in . Moreover, we examine the
relationships between the representations of orthogonal polynomials in terms of differential
or difference operators and our generating functions. The connections provide practical
criteria to determine when functions of a certain form are orthogonal polynomials.
2000 AMS Mathematics Subject Classification: 42C05, 46L53.
Key words and phrases: Interacting Fock space, probability measure, orthogonal
polynomial, SzegŲ-Jacobi parameters, pre-generating function, generating function,
multiplicative renormalization, differential and difference operators.