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Proceedings of the American Mathematical Society

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Inversion and representation for the Poisson-Laguerre transform

Author: Deborah Tepper Haimo
Journal: Proc. Amer. Math. Soc. 91 (1984), 559-567
MSC: Primary 44A15; Secondary 39A99
MathSciNet review: 746090
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Abstract: The Poisson-Laguerre transform of a function $ \phi $ is given by

$\displaystyle u(n,t) = \sum\limits_{m = 0}^\infty {g(n,m;t)\phi (m)\frac{{m!}}{{\Gamma (m + \alpha + 1)}}} $

where $ g$, defined by

$\displaystyle g(n,m;t) = \frac{{\Gamma (n + m + \alpha + 1)}}{{n!m!}}\frac{{{t^... ...ot _2}{F_1}\left( { - n, - m; - n - m - \alpha ;1 - \frac{1}{{{t^2}}}} \right),$

s the associated function of the source solution $ g(n;t) = g(n,0;t)$ of the Laguerre difference heat equation

$\displaystyle {\nabla _n}u(n,t) = {u_t}(n,t),$


$\displaystyle {\nabla _n}f(n) = (n + 1)f(n + 1) = (2n + \alpha + 1)f(n) + (n + \alpha )f(n - 1).$

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 $ (*)$.

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