The paper briefly introduces the Mehler-Fock transform of \(\mu \) th order ( \(\mu \) MFT) and its fundamental properties. Recurrence relation and estimate of derivative of associated Legendre function \(P_{-\frac{1}{2}+i\eta }^{-\mu }(\cdot )\) is obtained. It is shown that both the \(\mu \) MFT and pseudo-differential operator are continuous linear mappings between the test function spaces \(\mathcal {Q}_{\alpha }\) and \(\Pi _\alpha \) . The Mehler-Fock potential \(S_\mu ^{s}\) is defined over the space \(\mathcal {Q}_{\alpha }\) and some of its properties are discussed. It is proved that the Sobolev-type space \(W^{s,p}_\mu \big (\mathbb {I}\big )\) is complete w.r.t. the norm \(\Vert \cdot \Vert _{W^{s,p}_\mu }\) and the Mehler-Fock potential \(S_\mu ^{s}\) is an isometry of \(W^{s,p}_\mu \) .
