In this paper we consider a second-order singular differential operator Δα,nu=u′′+((2α+1)/x)u′-4n(α+n)x2u,α>-1/2,n∈N\documentclass[12pt]{minimal}
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\begin{document}$$ \Delta _{\alpha ,n}u = u'' + ((2\alpha +1)/x)u'-\frac{4n(\alpha +n)}{x^{2}}u, \alpha >-1/2, n\in {\mathbb {N}}$$\end{document} on the half line which generalize the Bessel operator Δαu=u′′+((2α+1)/x)u′,α>-1/2\documentclass[12pt]{minimal}
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\begin{document}$$ \Delta _{\alpha }u = u'' + ((2\alpha +1)/x)u', \alpha >-1/2 $$\end{document}. A generalized integral transform Tφα,n\documentclass[12pt]{minimal}
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\begin{document}$$ T_{\varphi }^{\alpha ,n}$$\end{document} associated with Δα,n\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _{\alpha ,n}$$\end{document} is studied in Lα,n2\documentclass[12pt]{minimal}
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\begin{document}$$ L_{\alpha ,n}^{2}$$\end{document} and we have established a relation between Tφα,n\documentclass[12pt]{minimal}
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\begin{document}$$ T_{\varphi }^{\alpha ,n}$$\end{document} its adjoint and generalized Bessel Plancherel transform ϕα,n\documentclass[12pt]{minimal}
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\begin{document}$$\phi _{\alpha ,n}$$\end{document} associated with Δα,n.\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _{\alpha ,n}.$$\end{document} We derive new results concerning the relation between ϕα,n\documentclass[12pt]{minimal}
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\begin{document}$$\phi _{\alpha ,n}$$\end{document}, and the generalized Riemann-Liouville transform, generalized Weyl transform, generalized Sonine transform and a Hankel potential type transform associated with Δα,n.\documentclass[12pt]{minimal}
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\begin{document}$$ \Delta _{\alpha ,n}.$$\end{document}
