The Mellin-Whittaker integral transform
The author gives an inversion formula for the integral transform \(\iint K(\xi,\eta,\alpha,\beta,\lambda)f(\xi,\eta,\lambda)d\xi d\eta =F(\alpha,\beta,\lambda)\) with the kernel \[ K=\{(2\lambda)^{2i\alpha +1}B(i(\alpha +\beta)+1/2,i(\alpha -\beta)+1/2)/_{2\Gamma (2i\alpha +1)}\}\cdot \] \[ \eta^{2i}e^{\beta \pi sign \xi \eta -i\lambda \xi \eta}\Phi (i(\alpha +\beta)+1/2,2i\alpha +1,2i\lambda \xi \eta), \] where \(\Phi\) is the degenerate hypergeometric function. The mentioned integral transform sends the Laplace-Beltrami operator \(\partial^ 2/\partial \xi \partial \eta +i\lambda (\xi \partial /\partial \xi -\eta \partial /\partial \eta)\) appearing in the theory of symmetrical spaces into a multiplication by a function.
scientific article; zbMATH DE number 4002768