Multiplicative renormalization and generating functions. I. (Q1811454)

Let \(\mu\) be a probability measure on \(\mathbb{R}\) with finite moments of all orders. Then there exists a sequence of polynomials \(P_{n}(x)\) which are orthogonal in \(L^{2}(\mu)\). In this paper the authors introduce a method of finding a generating function \(\psi(t,\,x)=\sum_{n=0}^{\infty}P_{n}(x)t^{n}\) for the sequence \(P_{n}(x)\). This function \(\psi(t,\,x)\) has the form \(\frac{\varphi(t,\,x)}{E_{x}\varphi(t,\,x)}\), where \(\varphi(t,\,x)\) is an appropriate function and \(E_{x}\) denotes the expectation in the \(x\)-variable with respect to the distribution \(\mu\). The construction of the function \(\varphi(t,\,x)\) is based on a result whose proof is given in a subsequent paper of the authors which is going to appear in the same journal. Several examples of known distributions are considered and the proposed method yields many sequences of classical orthogonal polynomials.