Laguerre random differential polynomials: definition, differential and statistical properties

[EN] In this paper we introduce the Laguerre polynomials as mean square solutions of random differential equations. The study is based on the construction of an infinite random power series solution which becomes a random polynomial under certain conditions to be satisfied by the single involved random coefficient, denoted by A. This approach allows us to introduce the concept of Laguerre polynomials associated to the random variable A retaining their deterministic definition when the probability mass of A is concentrated in a nonnegative integer. As a result, we provide a natural way to extend the deterministic Laguerre polynomials to the random framework. In addition, the main statistical functions of the approximate solution stochastic process obtained by truncation of the exact power series solution, which generates random Laguerre polynomials, are also given. Several illustrative examples are provided.