Four selfreciprocal integral transformations of Hankel type are defined through(ℋi,μf)(y)=Fi(y)=∫0∞αi(x)ℊi,μ(xy)f(x)dx, ℋi,μ−1=ℋi,μ,where i=1,2,3,4; μ≥0; α1(x)=x1+2μ, ℊ1,μ(x)=x−μJμ(x), Jμ(x) being the Bessel function of the first kind of order μ; α2(x)=x1−2μ, ℊ2,μ(x)=(−1)μx2μℊ1,μ(x); α3(x)=x−1−2μ, ℊ3,μ(x)=x1+2μℊ1,μ(x), and α4(x)=x−1+2μ, ℊ4,μ(x)=(−1)μxℊ1,μ(x). The simultaneous use of transformations ℋ1,μ, and ℋ2,μ, (which are denoted by ℋμ) allows us to solve many problems of Mathematical Physics involving the differential operator Δμ=D2+(1+2μ)x−1D, whereas the pair of transformations ℋ3,μ and ℋ4,μ, (which we express by ℋμ*) permits us to tackle those problems containing its adjoint operator Δμ*=D2−(1+2μ)x−1D+(1+2μ)x−2, no matter what the real value of μ be. These transformations are also investigated in a space of generalized functions according to the mixed Parseval equation∫0∞f(x)g(x)dx=∫0∞(ℋμf)(y)(ℋμ*g)(y)dy,which is now valid for all real μ.
