Lebedev's type index transforms with the squares of the associated Legendre functions

The classical Lebedev index transform (1967), involving squares and products of the Legendre functions is generalized on the associated Legendre functions of an arbitrary order. Mapping properties are investigated in the Lebesgue spaces. Inversion formulas are proved. As an interesting application, a solution to the boundary value problem for a third order partial differential equation is obtained.