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We propose the study of some questions related to the Dunkl-Hermite semigroup. Essentially, we characterize the images of the Dunkl-Hermite-Sobolev space, $𝒮\left(ℝ\right)$ and $L_{\alpha }^p\left(ℝ\right)$, $1<p<\infty$, under the Dunkl-Hermite semigroup. Also, we consider the image of the space of tempered distributions and we give Paley-Wiener type theorems for the transforms given by the Dunkl-Hermite semigroup.

In this work we consider the Dunkl operator on the complex plane, defined by $${𝒟}_{k}f\left(z\right)=\frac{d}{dz}f\left(z\right)+k\frac{f\left(z\right)-f(-z)}{z},k\ge 0.$$ We define a convolution product associated with ${𝒟}_k$ denoted ${*}_k$ and we study the integro-differential-difference equations of the type $\mu {*}_{k}f={\sum }_{n=0}^{\infty }{a}_{n,k}{𝒟}_k^{n}f$, where $\left(a_{n,k}\right)$ 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.