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Integro-dierential-dierence equations associated with the Dunkl operator and entire functions

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N´ ejib Ben Salem, Samir Kallel

Integro-dierential-dierence equations associated with the Dunkl operator and entire functions

Comment.Math.Univ.Carolinae 45,4 (2004) 699-725.

Abstract: In this work we consider the Dunkl operator on the complex plane, defined by

Dkf(z) = d

dzf(z) +kf(z)−f(−z) z , k≥0.

We define a convolution product associated withDk denoted ∗k and we study the integro-differential-difference equations of the typeµ∗kf =P∞

n=0an,kDknf, where (an,k) is a sequence of complex numbers andµis a measure over the real line. We show that many of these equations provide representations for particular classes of entire functions of exponential type.

Keywords: Dunkl operator, Fourier-Dunkl transform, entire function of exponen- tial type, integro-differential-difference equation

AMS Subject Classification: 30D15, 33E30, 34K99, 44A35, 45J05

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