Sonine Transform Associated to the Bessel-Struve Operator

In this paper we consider the Bessel-Struve operator $l_{\alpha }$ and the Bessel-Struve intertwining operator ${\chi }_{\alpha }$ and its dual, we define and study the Bessel-Struve Sonine transform $S_{\alpha ,\beta }$ on $\mathcal{E}(\mathbb{R})$. We prove that $S_{\alpha ,\beta }$ is a transmutation operator from $l_{\alpha }$ into $l_{\beta }$ on $\mathcal{E}(\mathbb{R})$ and we deduce similar result for its dual $S_{\alpha ,\beta }^{\ast }$ on ${\mathcal{E}}^{\mathrm{\prime }}(\mathbb{R})$. Furthermore, invoking Weyl integral transform and the Dual Sonine transform ${}^t{S}_{\alpha ,\beta }$ on $\mathcal{D}(\mathbb{R})$, we get a relation between the Bessel-Struve transforms ${\mathcal{F}}_{BS}^{\alpha }$ and ${\mathcal{F}}_{BS}^{\beta }$.