Using the q2-Laplace transform and its inverse transform introduced early on by Hahn and deeply studied by Abdi we have prove that the q-analogues of the heat and wave equations are linked as in the classical case of Bragg and Dettman. As an application, we proved first, through the q-wave polynomials, that the q-Hermite and the q-little Jacobi polynomials are related. Second, we have given a q-analogue of the Poisson kernel studied by Fitouhi and Annabi.
AbdiWH. Certain inversion and representation formulas for the q-Laplace transforms, Math Zeitschr1964; 83: 238–249.
6.
GasperGRahmanM. Basic Hypergeometric Series (Encyclopedia of Mathematics and Its Applications, vol. 35). Cambridge: Cambridge University Press, 1990.
7.
KacVGCheeungP. Quantum Calculus (Universitext). Berlin: Springer-Verlag, New York, 2002.
8.
JacksonFH. On a q-definite integrals. Q J Pure Appl Math1910; 41: 193–203.
9.
De SolaAKacVG. On integral representations of q-gamma and q-beta functions. Preprint arXiv:math.QA/0302032.
10.
FitouhiABettaibiNBrahimK. The Mellin transform in quantum caculus. Constr Approx2005; 16: 305–323.
11.
KoornwinderTom H. q-Special Function, A Tutorial. Mathematical Preprint Series Report 94–08, University of Amsterdam, The Netherlands, 1994.
12.
FitouhiAHamzaMBouzeffourF. The q-jα Bessel function. J Approx Theory2002; 115: 114–116.
13.
KoornwinderTom H. Special function and q-commuting variable. Special Function q-series and Related Topics (Toronto, ON, 1995), Fields Inst Commun, vol. 14. Providence, RI: American Mathematical Society, 1997, pp. 131–166.
14.
BouzeffourF. q-élément d’analyse. These, Faculté des Sciences de Tunis, Octobre2002.
15.
MoakDS. The q-analogue of the Laguerre polynomials. J Math Anal Appl1981; 81: 21–47.
16.
FitouhiAAnnabiH. La g-fonction de Littlewood–Paley associée à un operateur differentiels sur (0,∞) contenant l’operateur de Bessel. C R Acad Sci Paris Ser11986; 303: 411–413.
17.
IsmailMEH. A simple proof of Ramanujan’s sum. Proc Amer Math Soc1977; 63: 185–186.
18.
BooleG. A Treatise on the Calculus of Finite Differences, 2nd edn.New York: MacMillan and Company, 1872.
