Sonine Transform Associated to the Bessel-Struve Operator

TL;DR

This paper introduces and analyzes the Bessel-Struve Sonine transform, demonstrating its role as a transmutation operator between Bessel-Struve operators and establishing relations between associated transforms.

Contribution

It defines the Bessel-Struve Sonine transform and proves its properties as a transmutation operator, extending the understanding of Bessel-Struve related integral transforms.

Findings

S_{eta,eta} is a transmutation operator from l_eta to itself

S_{eta,eta}^* acts on distributions in \\mathcal{E}'(\\mathbb{R})

Relations between Bessel-Struve transforms \\mathcal{F}^\alpha_{BS} and \\mathcal{F}^\beta_{BS} are established.

Abstract

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 $S_{\alpha,\beta}^*$ on $\mathcal{E}'(\mathbb{R})$. Furthermore, invoking Weyl integral transform and the Dual Sonine transform $^tS_{\alpha,\beta}$ on $\mathcal{D}(\mathbb{R})$, we get a relation between the Bessel-Struve transforms $\mathcal{F}^\alpha_{BS} $ and $\mathcal{F}^\beta_{BS} $.