Sonine transform associated with the Bessel-Struve kernel (Q2923133)

The paper deals with the Sonine integral and other related transforms. The Sonine integral transform \(S_{\alpha, \beta}\) of a function \(f\) is defined as NEWLINE\[NEWLINE S_{\alpha, \beta}(f)(x)=\frac{ 2\Gamma (\alpha +1)}{\Gamma (\beta +1)\Gamma (\alpha -\beta)}\int_0^1 (1-r^2)^{\alpha -\beta -1}f(rx)r^{2\beta +1} dr,\quad 0\leq x .NEWLINE\]NEWLINE Consider the differential operator NEWLINE\[NEWLINE\ell_\alpha u(x)=\frac{d^2 u}{dx^2}+\frac{2\alpha +1}{x}\left( \frac{du}{dx}-\frac{du}{dx}(0)\right),NEWLINE\]NEWLINE and the Bessel-Struve intertwining integral operator \(\chi_\alpha\) defined by NEWLINE\[NEWLINE \chi_\alpha(f)(x)=\frac{ 2\Gamma (\alpha +1)}{\sqrt{\pi}\Gamma (\alpha +1/2)}\int_0^1 (1-t^2)^{\alpha -1/2}f(xt)dt.NEWLINE\]NEWLINE The solution of the differential equation NEWLINE\[NEWLINE\ell_\alpha u=\lambda^2 u,\; \text{ with }\quad u(0)=1, \quad u'(0)=\frac{\lambda \Gamma(\alpha +1)}{\sqrt{\pi}\Gamma (\alpha +3/2)},NEWLINE\]NEWLINE is the Bessel-Struve function \(\Phi_\alpha\) given by NEWLINE\[NEWLINE\Phi_\alpha (\lambda x)=\frac{ 2\Gamma (\alpha +1)}{\sqrt{\pi}\Gamma (\alpha +1/2)}\int_0^1 (1-t^2)^{\alpha -1/2}e^{\lambda xt}dt.NEWLINE\]NEWLINE It is shown that NEWLINE\[NEWLINE S_{\alpha, \beta } \left( \Phi_\beta(\lambda )\right)(x)=\Phi_\alpha (\lambda x), NEWLINE\]NEWLINE and for \( f\in C^2\), NEWLINE\[NEWLINE \ell_\alpha\left( S_{\alpha, \beta } (f)\right) (x)=S_{\alpha, \beta } \left( \ell_\beta (f)\right)(x).NEWLINE\]NEWLINE Furthermore, it is shown that \(S_{\alpha, \beta}\) is a transmutation operator in the sense that \(S_{\alpha , \beta}\) is a topological isomorphism from \({\mathcal E},\) the space of all infinitely differentiable functions with its standard topology into itself, and NEWLINE\[NEWLINE S_{\alpha , \beta}=\chi_\alpha \circ \chi^{-1}_\beta, \quad \text{and } S^{-1}_{\alpha , \beta}=\chi_\beta \circ \chi^{-1}_\alpha. NEWLINE\]NEWLINE Passing from the space \({\mathcal E}\) to its dual space \({\mathcal E}',\) the authors define the Bessel-Struve transform of \(T\in {\mathcal E}'\) as NEWLINE\[NEWLINE {\mathcal F}^\alpha_{BS} (T)(\lambda)=\langle T, \Phi_\alpha (-i\lambda)\rangle,NEWLINE\]NEWLINE and it is shown that \( {\mathcal F}^\alpha_{BS} (T)={\mathcal F}^\beta_{BS} \left( S_{\alpha,\beta}^*T\right),\) where \(S_{\alpha,\beta}^*\) is the conjugate operator. Other relations regarding the transforms of the convolution are obtained.