Duhamel convolution product in the setting of Quantum calculus

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.