Two Hankel transformations called Hankel-Clifford transformations are defined in certain spaces of distributions by means of adjoint operators. In fact, the first Hankel-Clifford transformation is defined as the adjoint of the second Hankel-Clifford transformation, and conversely. These transforms are applied in solving some distributional partial differential equations of Kepinsky type. For \(I=(0,\infty)\) and \(\mu\geq 0\), testing function spaces \(S(I)\) and \(H_ \mu(I)\) are defined (details are omitted).
The classical Hankel-Clifford transformations \(h_{1,\mu}\) and \(h_{2,\mu}\) are respectively defined as: \[ \begin{aligned} (h_{1,\mu}\phi)(y) &= \Phi(y)=y^ \mu \int_ 0^ \infty C_ \mu(xy)\phi(x)dx,\tag{1} \\ (h_{2,\mu}\psi)(y) &= \Psi(y)=\int_ 0^ \infty x^ \mu C_ \mu(xy)\psi(x)dx,\tag{2} \\ \end{aligned} \] for all \(\phi(x)\in H_ \mu(I)\) and \(\psi(x)\in S(I)\); \[ C_ \mu(x)=\sum_{r=0}^ \infty {{(-1)^ r x^ r} \over {r!\Gamma(\mu+r+1)}}. \] The main theorems proved are as follows:
Theorem 1: The first Hankel-Clifford transformation \(h_{1,\mu}\) is an automorphism on \(H_ \mu(I)\).
Theorem 2: The second Hankel-Clifford transformation \(h_{2,\mu}\) is an automorphism on \(S(I)\).
The generalized Hankel-Clifford transformation \(h'_{1,\mu}\) on \(S'(I)\) is defined as: \[ \langle h'_{1,\mu}f,\phi\rangle = \langle f,h_{2,\mu}\phi\rangle, \] for all \(f\in S'(I)\) and \(\phi\in S(I)\).
Analogously, the second generalized Hankel-Clifford transformation \(h'_{2,\mu}\) on \(H'_ \mu(I)\) is defined as \[ \langle h'_{2,\mu}f,\phi\rangle = \langle f,h_{1,\mu}\phi\rangle, \] for all \(f\in H'_ \mu(I)\) and \(\phi\in H_ \mu(I)\).
It is respectively established that \(h'_{1,\mu}\) is an automorphism on \(S'(I)\) and \(h'_{2,\mu}\) is an automorphism on \(H'_ \mu(U)\).
Various interesting results are also proved in the sequel. Operation transform formulae are also obtained and applied in solving partial differential equations of Kepinksi type such as: \[ x{{\partial^ 2u}\over{\partial x^ 2}}+{(1+\mu){\partial u}\over{\partial x}} - {\lambda{\partial u} \over {\partial t}}=0;\qquad \mu\geq 0,\;\lambda>0,\;0<t<\infty,\;0<x<\infty, \] with the initial condition \(\mu(x,t)\to f(x)\in H'_ \mu(I)\) as \(t\to 0^ +\).