The fundamental importance of differential equations with three singularities in Mathematical Statistics

Abstract

It is well-known that the solution of a second order linear differential equation with at most five singularities plays a fundamental role in Mathematical Physics. In this paper it is shown that this statement also applies to Mathematical Statistics but with the difference that an equation with three singularities will suffice. Two wide classes of probability distributions are defined as solutions of such a differential equation, one for continuous distributions and one for discrete distributions. These two classes contain as members all the distributions which are normally considered as of importance in Mathematical Statistics. In the continuous case the probability functions are solutions of the relevant second order equation, while in the discrete case the probability generating functions are solutions there-of. By defining appropriate multidimentional extensions corresponding differential equations are obtained for continuous and discrete multivariate distributions.