Abstract: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.
Keywords: Dunkl operator, Fourier-Dunkl transform, entire function of exponential type, integro-differential-difference equation
AMS Subject Classification: 30D15, 33E30, 34K99, 44A35, 45J05