One generalization of the classical moment problem

Let $ast_P$ be a product on $l_{rm{fin}}$ (a space of all finite sequences) associated with a fixed family $(P_n)_{n=0}^{infty}$ of real polynomials on $mathbb{R}$. In this article, using methods from the theory of generalized eigenvector expansion, we investigate moment-type properties of $ast_P$-positive functionals on $l_{rm{fin}}$. If $(P_n)_{n=0}^{infty}$ is a family of the Newton polynomials $P_n(x)=prod_{i=0}^{n-1}(x-i)$ then the corresponding product $star=ast_P$ is an analog of the so-called Kondratiev--Kuna convolution on a Fock space. We get an explicit expression for the product $star$ and establish a connection between $star$-positive functionals on $l_{rm{fin}}$ and a one-dimensional analog of the Bogoliubov generating functionals (the classical Bogoliubov functionals are defined correlation functions for statistical mechanics systems).