Deformations and q-Convolutions. Old and New Results
Marek Bożejko, Wojciech Bożejko
Abstract
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 $$\mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> . 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 ( $$q=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ) convolutions, the case $$q=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , 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.