Bivariate delta-evolution equations and convolution polynomials: Computing polynomial expansions of solutions

This paper describes an application of Rota and collaborator’s ideas on the foundations on combinatorial theory to the computing of solutions of some linear functional partial differential equations. We give a dynamical interpretation of the convolution families of polynomials as the entries, in the matrix representation, of the exponentials of certain contractive linear operators in the ring of formal power series. This is the starting point to get symbolic solutions to some functional-partial differential equations. We introduce the bivariate convolution product of convolution families to obtain symbolic solutions of a natural extension of a functional-evolution equation related to delta-operators. We put some examples to show how these symbolic methods allow us to get closed formulas for solutions of genuine partial differential equations and of some functional-evolution equations. We create an adequate framework to base, theoretically, some of the constructions and to get some existence and uniqueness results. This paper is dedicated to Jose Maria Montesinos Amilibia with admiration and on occasion of his 65th birthday.