Deformations and q-convolutions. Old and new results

2023/11/10 by Marek Bożejko, Bozejko, Marek, Wojciech Bożejko +1
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2311.06405

openalex publication_date 2023/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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. 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.

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