International Journal of Mathematics and Mathematical Sciences
Volume 15 (1992), Issue 2, Pages 323-332
doi:10.1155/S0161171292000401

Hankel complementary integral transformations of arbitrary order

M. Linares Linares1 and J.M.R. Mendez Pérez2

1Departamento de Informática y Sistemas, Universidad de Las Palmas, Canary Islands, Las Palmas de Gran Canaria, Spain
 2Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de La Laguna, Tenerife, Canary Islands, La Laguna, Spain

Abstract

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.