Theory and algorithms of the Laguerre transform, Part I: Theory

The Laguerre transform, introduced by Keilson and Nunn and Sumita, provides an algorithmic framework for the computer evaluation of repeated combinations of continuum operations such as convolution, integration, differentiation and multiplication by polynomials. The procedure enables one to numerically evaluate many distribution results of interest, which have been available only formally behind the ‘Laplacian curtain’. Since the initial development, the formalism has been extended to incorporate matrix and bivariate functions and finite signed measures. The purpose of this paper is to summarize theoretical results on the Laguerre transform obtained up to date. In a sequel to this paper, a summary is given focusing on algorithmic aspects. The two summary papers will enable the reader to use the Laguerre transform with ease.