  We consider the Dunkl intertwining operator $V_\alpha$ and its dual
${}^tV_\alpha$, we define and study the Dunkl Sonine operator and its dual on
$\mathbb{R}$. Next, we introduce complex powers of the Dunkl Laplacian
$\Delta_\alpha$ and establish inversion formulas for the Dunkl Sonine operator
$S_{\alpha,\beta}$ and its dual ${}^tS_{\alpha,\beta}$. Also, we give a
Plancherel formula for the operator ${}^tS_{\alpha,\beta}$.
