Inversion of the Dunkl-Hermite Semigroup

Let {e^-c^Ӄ^α/ =xc ≥ 0} be the Dunkl-Hermite semigroup on the real line R, defined by [e^-c^Ӄ^α f ](x) = ʃ_ R Ӄ^ α_ c (x,₰) f (₰ )dμα (₰ ) ; x ϵ R ; where Ӄ^ α_ c (x,₰ ) =∑^∞_ n=0 e^-cn H^a_ n (x)Ha n (₰ ). Here, H^a_ n ;n ϵ N, are the Dunkl-Hermite polynomials which are the eigenfunctions of the operator D^2_ α -2xd/dx, D_α being the Dunkl operator on the real line. For ₰c 0, we give a representation for inverting the semigroup. Next, we extend e􀀀cH a and we give an integral representation of it for ₰c 0. Moreover, in this last case, we characterize the domain in which e^-cӃ^ a is well defined.