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 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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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     author = {Salem, N\'ejib Ben and Kallel, Samir},
     title = {Integro-differential-difference equations associated with the {Dunkl} operator and entire functions},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {699--725},
     year = {2004},
     volume = {45},
     number = {4},
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     zbl = {1098.30019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2004_45_4_a10/}
}
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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/