In this paper we introduce some differential and integral operators and study the continuity mapping on test function spaces \(\mathbb {H}_\mu (\mathbb {I})\) and \(\mathbb {S}_\mu (\mathbb {I})\) . Further, the operational calculus of differential, integral operators as well as the \(\mu \) th order Mehler–Fock transform have also been discussed. The translation and convolution operators are introduced and obtain some their estimates. Finally, this transform is used in the solution of some partial differential equations of Kepinski type.
