The reduced dual Poisson-Laguerre transform (Q1058096)

Let \(g_{\alpha}(x;t)=g_{\alpha}(x,0;t)\) denote the fundamental solution of the differential Laguerre heat equation, and let \(d\Lambda_{\alpha}(x)=(1/\Gamma (\alpha +1))x^{\alpha} e^{-x} dx.\) The author considers the transform \[ (*)\quad f(t)=\int^{\infty}_{0}g_{\alpha}(x;t)\phi (x)d\Lambda_{\alpha}(x),\quad t>0, \] from the point of view that, within the region of convergence of the integral, the transform may be looked upon as a dual Laguerre temperature at \(y=0,t\) seconds after the initial dual Laguerre temperature was \(\phi\) (y). An inversion algorithm is derived, and a criterion for the representation of a function as a dual Laguerre transform (*) is established. The transform (*) is a Laplace transform: the author derives an inversion formula corresponding to this reliance.