The Ramanujan Journal

, Volume 28, Issue 3, pp 443-461

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

The q-cosine Fourier transform and the q-heat equation

  • Ahmed FitouhiAffiliated withDepartment of Mathematics, Faculty of Sciences of Tunis University El-Manar Email author 
  • , Fethi BouzeffourAffiliated withDepartment of Mathematics, College of Sciences, King Saud University


The aim of this work is to establish in great detail The q-Fourier analysis related to the q-cosine. The wise reader will note that the considered q-cosine coincides with the one given by T.H. Koornwinder and S.F. Swarttouw. Through the q-cosine product formula, we define and analyze the properties of the q-even translation and the q-convolution. Adopting the Titchmarsh approach, we study the q-cosine Fourier transform and its inverse formula.

The second theme of this paper is an application of the q-Fourier analysis developed earlier. We extend the heat representation theory inaugurated by P.C. Rosenbloom and D.V. Widder to the q-analogue. We construct the q-solution source, the q-heat polynomials and solve the q-analytic Cauchy problem.


Basic orthogonal polynomials and functions Basic hypergeometric integrals

Mathematics Subject Classification (2000)

33D45 33D6043