Appears in collection : Combinatorics and Arithmetic for Physics - 2024

This talk is dedicated to 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 related 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. Keywords — Ortogonal polynomials, Measures convolution, Khintchine inequality, q-Gaussian operators. For more details, see: Marek Bozejko, Wojciech Bozejko, (dedicated to Professor Jan Stochel on the occasion of his 70th birthday). Deformations and q-Convolutions. Old and New Results, Complex Analysis and Operator Theory (2024). https://link.springer.com/article/10.1007/s11785-024-01572-8