Duhamel convolution product in the setting of Quantum calculus

TL;DR

This paper extends classical analysis by introducing $q$-analogues of the Duhamel product and integration operator within quantum calculus, demonstrating their algebraic properties on Wiener algebra functions.

Contribution

It defines $q$-Duhamel product and $q$-integration operator, proving the Wiener algebra forms a Banach algebra under these operations and characterizing their algebraic features.

Findings

Wiener algebra is a Banach algebra under $q$-Duhamel product

Cyclic vectors of the $q$-integration operator are described

The commutant of the $q$-integration operator is characterized

Abstract

In this paper we introduce the notions of $q$-Duhamel product and $q$-integration operator. We prove that the classical Wiener algebra $W(\mathbb{D})$ of all analytic functions on the unit disc $\mathbb{D}$ of the complex plane $\mathbb{C}$ with absolutely convergent Taylor series is a Banach algebra with respect to $q$-Duhamel product. We also describe the cyclic vectors of the $q$-integration operator on $W(\mathbb{D})$ and characterize its commutant in terms of the $q$-Duhamel product operators.