Explicit Formulas for the One-Parameter Group Generated by the Dunkl Operator on $\mathbb{R}$

Let $T_b$ be the Dunkl operator for the reflection group $G=\mathbb{Z}\mathrm{/}2\mathbb{Z}$, and $D_b:=\mathrm{\mid }x{\mathrm{\mid }}^b{T}_b\mathrm{\mid }x{\mathrm{\mid }}^{-b}$. We compute explicitly the unitary one-parameter group $e^{t{D}_b}$ generated by $D_b$. We obtain two representations: a boundary value representation from the upper and lower half-planes, and a real-variable formula consisting of a translation term and a principal value integral term with an explicit kernel expressed in terms of Legendre functions.