Deformations and q-Convolutions. Old and New Results

This paper is the survey of some of our results related to q -deformations of the Fock spaces and related to q -convolutions for probability measures on the real line R . The main idea is done by the combinatorics of moments of the measures and related q -cumulants of different types. The main and interesting q -convolutions are related to classical continuous (discrete) q -Hermite polynomial. Among them are classical ( q = 1 ) convolutions, the case q = 0 , gives the free and Boolean relations, and the new class of q -analogue of classical convolutions done by Carnovole, Koornwinder, Biane, Anshelovich, and Kula. The paper contains many questions and problems related to the positivity of that class of q -convolutions. The main result is the construction of Brownian motion related to q -Discrete Hermite polynomial of type I.