Geodesic
Integro-differential-difference equations associated with the Dunkl operator and entire functions
Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 4, pp. 699-725

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In this work we consider the Dunkl operator on the complex plane, defined by $$ \Cal D_k f(z)=\frac{d}{dz}f(z)+k\frac{f(z)-f(-z)}{z}, k\geq 0. $$ We define a convolution product associated with $\Cal D_k$ denoted $\ast_k$ and we study the integro-differential-difference equations of the type $\mu \ast_k f=\sum_{n=0}^{\infty}a_{n,k}\Cal D^n_k f$, where $(a_{n,k})$ is a sequence of complex numbers and $\mu $ is a measure over the real line. We show that many of these equations provide representations for particular classes of entire functions of exponential type.
Classification : 30D05, 30D15, 33E30, 34K99, 34M05, 44A35, 45J05
Keywords: Dunkl operator; Fourier-Dunkl transform; entire function of exponential type; integro-differential-difference equation 
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     title = {Integro-differential-difference equations associated with the {Dunkl} operator and entire functions},
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Salem, Néjib Ben; Kallel, Samir. Integro-differential-difference equations associated with the Dunkl operator and entire functions. Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 4, pp. 699-725. http://geodesic.mathdoc.fr/item/CMUC_2004__45_4_a10/