Sonine Transform Associated to the Bessel-Struve Operator

Abstract

In this paper we consider the Bessel-Struve operator $l_α$ and the Bessel-Struve intertwining operator $χ_α$ and its dual, we define and study the Bessel-Struve Sonine transform $S_{α,β}$ on $\mathcal{E}(\mathbb{R})$. We prove that $S_{α,β}$ is a transmutation operator from $l_α$ into $l_β$ on $\mathcal{E}(\mathbb{R})$ and we deduce similar result for its dual $S_{α,β}^*$ on $\mathcal{E}'(\mathbb{R})$. Furthermore, invoking Weyl integral transform and the Dual Sonine transform $^tS_{α,β}$ on $\mathcal{D}(\mathbb{R})$, we get a relation between the Bessel-Struve transforms $\mathcal{F}^α_{BS} $ and $\mathcal{F}^β_{BS} $.