On the generalized Hankel-Clifford transformation of arbitrary order
The generalized Hankel-Clifford integral transformations of order \((\alpha-\beta)\geq -1/2\) are defined by \[ (h_{1,\alpha,\beta}f)(y)= y^{-\alpha-\beta} \int^\infty_0 C_{\alpha,\beta}(xy) f(x) dx,\tag{1} \] \[ (h_{2,\alpha,\beta}f)(y)= \int^\infty_0 x^{-\alpha-\beta} C_{\alpha,\beta}(xy) f(x) dx,\tag{2} \] where \(C_{\alpha,\beta}(xy)= (xy)^{(\alpha+ \beta)/2}J_{\alpha- \beta}(2\sqrt{xy})\) is a generalized Bessel-Clifford function of first kind of order \((\alpha- \beta)\). The above two transformations are simultaneously investigated in certain spaces of distributions using the procedure of the adjoint operator for which a Parseval relation involving (1) and (2) is exploited so as to define the first distributional transformation as the adjoint operator of the second one, and conversely for \((\alpha- \beta)\geq -1/2\) as well as for negative real values of order \((\alpha- \beta)\).
generalized Hankel-Clifford integral transformations
scientific article; zbMATH DE number 1525092