Let *P be a product on lfin (a space of all finite sequences) associated with a fixed family <$E (P sub n ) sub n=0 sup inf> of real polynomials on R. In this article, using methods from the theory of generalized eigenvector expansion, we investigate moment-type properties of *P-positive functionals on lfin. If <$E (P sub n ) sub n=0 sup inf> is a family of the Newton polynomials <$E P sub n (x)~=~PI sub i=0 sup n-1 (x~-~i)> then the corresponding product * = *P is an analog of the so-called Kondratiev - Kuna convolution on a "Fock space". We get an explicit expression for the product * and establish a connection between *-positive functionals on lfin and a one-dimensional analog of the Bogoliubov generating functionals (the classical Bogoliubov functionals are defined correlation functions for statistical mechanics systems).

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