The ring of T-covariants

Starting from the invariant theory of binary forms, we extend the classical notion of covariants and introduce the ring of T-covariants. This ring consists of maps defined on a ring of polynomials in one variable which commute with all translation operators. We study this ring and we show some of its meaningful features. We state an analogue of the classical Hermite reciprocity law, and recover the Hilbert series associated with a suitable double grading via elementary theory of partitions. Together with classical covariants of binary forms other remarkable mathematical notions, such as orthogonal polynomials and cumulants, turn out to have a natural and simple interpretation in this algebraic framework. As a consequence, a Heine integral representation for the cumulants of a random variable is obtained.