On the q-translation associated with the Askey-Wilson operator

Bouzaffour

Description

In this article, we solve the open problem 24.5.6 given in the study of Ismail, which consists of extending the action of q -translation operators introduced by Ismail to some measurable functions by means of basic Fourier theory. Also, we prove that the q -exponential function is the only solution of the q -analogue of the Cauchy functional equation. As application we give an inversion formula for the q -Gauss Weierstrass transform. 2000 Mathematics Subject Classification: 33D45; 33D60.

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• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 9
• Langue: English

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BouzaffourAdvances in Difference Equations2012,2012:87 http://www.advancesindifferenceequations.com/content/2012/1/87

R E S E A R C HOpen Access On theqtranslation associated with the Askey Wilson operator Fethi Bouzaffour

Correspondence: fbouzaffour@ksu. edu.sa Department of Mathematics, College of Sciences, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia

Abstract In this article, we solve the open problem 24.5.6 given in the study of Ismail, which consists of extending the action ofqtranslation operators introduced by Ismail to some measurable functions by means of basic Fourier theory. Also, we prove that theqexponential function is the only solution of theqanalogue of the Cauchy functional equation. As application we give an inversion formula for theqGauss Weierstrass transform. 2000 Mathematics Subject Classification:33D45; 33D60. Keywords:basic orthogonal polynomials and functions, basic hypergeometric integrals

Introduction y The concept of theqtranslation operators E introduced by Ismail [1] was defined in q polynomials through their action on the continuousqHermite polynomialsH m (x | q) as follows n  y (q;q) 2 2 n (m−n)/4 E q H n (x|q) =H m (x|q)g n−m (y)q, (1) (q;q) (q;q) m n−m m=0 where 2 n/4 2iθ−inθ2−n2iθ2 g n (cos(θ)) =q(1 +e)e(−q e;q) n−1 . In others words 2 2 2 n/4j+(m+(n−m−2j) )/4 q y  q H m (x|q)H n−m−2j (y|q) E q H n (x|q,) = 2 2 (q;q(q;q) ) n 0≤m,j,m+2j≤n j (q;q) (q;q) m n−m−2j

where the polynomialsH n (x | q) are defined by (see [2,3]) n  (q;q) n i (n−2k)θ H n (cosθ|q) =e. (q;q) (q;q) n−k k k=0

(2)

y It wase with the Askey shown in [1] that theqtranslation operators E q commut Wilson operator D q on the space of all polynomials and by the use of the following expansion [3]

© 2012 Bouzaffour; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.