The Ramanujan Journal

, Volume 46, Issue 2, pp 345–356 | Cite as

Duhamel convolution product in the setting of quantum calculus

  • F. Bouzeffour
  • M. T. Garayev


In this paper, we introduce the notions of the 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 extended to the boundary is a Banach algebra with respect to the 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.


Duhamel product q-Difference operator q-Integral q-special functions q-Duhamel product 

Mathematics Subject Classification

Primary 33D45 Secondary 96J15 



The authors thank the referee for his valuable remarks and suggestions, which improved the paper.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesKing Saud UniversityRiyadhSaudi Arabia

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