Deformations and q-Convolutions. Old and New Results

Abstract

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\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document}. 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\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=1$$\\end{document}) convolutions, the case q=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=0$$\\end{document}, 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.