Inversion and representation for the Poisson-Laguerre transform

Abstract:

The Poisson-Laguerre transform of a function $\phi$ is given by \[ u(n,t) = \sum \limits _{m = 0}^\infty {g(n,m;t)\phi (m)\frac {{m!}}{{\Gamma (m + \alpha + 1)}}} \] where $g$, defined by \[ g(n,m;t) = \frac {{\Gamma (n + m + \alpha + 1)}}{{n!m!}}\frac {{{t^{m + m}}}}{{{{(1 + t)}^{n + m + \alpha + 1}}}}{ \cdot _2}{F_1}\left ( { - n, - m; - n - m - \alpha ;1 - \frac {1}{{{t^2}}}} \right ),\] s the associated function of the source solution $g(n;t) = g(n,0;t)$ of the Laguerre difference heat equation \[ {\nabla _n}u(n,t) = {u_t}(n,t),\] with \[ {\nabla _n}f(n) = (n + 1)f(n + 1) = (2n + \alpha + 1)f(n) + (n + \alpha )f(n - 1).\] A simple algorithm for the inversion of the transform $(*)$ is derived. For $m = 0$, the transform $(*)$ is basically a power series so that the inversion algorithm leads to a useful identity involving binomial coefficients. In addition, a subclass of functions is characterized that is representable by a Poisson-Laguerre transform $(*)$.