International Journal of Mathematics and Mathematical SciencesVolume 15 (1992), Issue 2, Pages 323-332http://dx.doi.org/10.1155/S0161171292000401

## Hankel complementary integral transformations of arbitrary order

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

Received 13 November 1990; Revised 18 June 1991

Copyright © 1992 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Four selfreciprocal integral transformations of Hankel type are defined through(i,μf)(y)=Fi(y)=0αi(x)i,μ(xy)f(x)dx,i,μ1=i,μ,where i=1,2,3,4; μ0; α1(x)=x1+2μ, 1,μ(x)=xμJμ(x), Jμ(x) being the Bessel function of the first kind of order μ; α2(x)=x12μ, 2,μ(x)=(1)μx2μ1,μ(x); α3(x)=x12μ, 3,μ(x)=x1+2μ1,μ(x), and α4(x)=x1+2μ, 4,μ(x)=(1)μx1,μ(x). The simultaneous use of transformations 1,μ, and 2,μ, (which are denoted by μ) allows us to solve many problems of Mathematical Physics involving the differential operator Δμ=D2+(1+2μ)x1D, whereas the pair of transformations 3,μ and 4,μ, (which we express by μ*) permits us to tackle those problems containing its adjoint operator Δμ*=D2(1+2μ)x1D+(1+2μ)x2, 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 equation0f(x)g(x)dx=0(μf)(y)(μ*g)(y)dy,which is now valid for all real μ.