Abel Expansions and Generalized Abel Polynomials in Stochastic Models

To build expansions, the family of the Abel polynomials $$ \left\{ {\left( {x - a} \right){{\left( {x - a - bn} \right)}^{n - 1}}/n!{\text{; n}} \in \mathbb{N}} \right\} $$ can be used as a basis in place of the classical family of monomials $$\left\{ {{x^n}/n!{\text{; }}n \in \mathbb{N}} \right\}$$. In that case we get Abel’s expansions that generalize Taylor’s ones. The purpose of the present paper is to show that these polynomials and expansions are present implicitely in several probability models, and that making explicit their hidden algebraic structure is very useful. More complex stochastic models can then also be considered, after extending the Abelian structure to more general polynomials.