A Class of Integral Operators and Generalized Bessel Plancherel Transform on $$L_{\alpha ,n}^{2}$$ L α , n 2

Abstract

In this paper we consider a second-order singular differential operator $$ \Delta _{\alpha ,n}u = u'' + ((2\alpha +1)/x)u'-\frac{4n(\alpha +n)}{x^{2}}u, \alpha >-1/2, n\in {\mathbb {N}}$$ on the half line which generalize the Bessel operator $$ \Delta _{\alpha }u = u'' + ((2\alpha +1)/x)u', \alpha >-1/2 $$ . A generalized integral transform $$ T_{\varphi }^{\alpha ,n}$$ associated with $$\Delta _{\alpha ,n}$$ is studied in $$ L_{\alpha ,n}^{2}$$ and we have established a relation between $$ T_{\varphi }^{\alpha ,n}$$ its adjoint and generalized Bessel Plancherel transform $$\phi _{\alpha ,n}$$ associated with $$\Delta _{\alpha ,n}.$$ We derive new results concerning the relation between $$\phi _{\alpha ,n}$$ , and the generalized Riemann-Liouville transform, generalized Weyl transform, generalized Sonine transform and a Hankel potential type transform associated with $$ \Delta _{\alpha ,n}.$$