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In this paper four self-reciprocal integral transformations of Hankel type are defined. The simultaneous use of trans-formations H1,α,β and H2,α,β (which are denoted by Hα,β) allows us to solve many problems of Mathematical Physics involving the differential operator ∆α,β= D2+4αx−1D, whereas the pair of transformations H3,α,β and H4,α,β (which we express by Hα,β) permits us to tackle those problems containing its adjoint operator, no matter what the real value of α − β be. These transformations are also investigated in a space of generalized functions according to the mixed Parseval equation.