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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 aPlancherel formula for the operator ${}^tS_{\alpha,\beta}$.

Sonine Transform Associated to the Dunkl Kernel on the Real Line