Abstract

Starting from the invariant theory of binary forms, we extend the classical notion of covariants and introduce the ring of $\mathcal {T}$ -covariants. This ring consists of maps defined on a ring of polynomials in one variable which commute with all the 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 the 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.

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.