A generating operator for Rankin-Cohen brackets

TL;DR

This paper introduces a new generating operator for Rankin-Cohen brackets, utilizing higher-dimensional residue calculus and representation theory to deepen understanding of differential operators between manifolds.

Contribution

It presents the first explicit formula for generating operators of Rankin-Cohen brackets, connecting residue calculus with representation theory.

Findings

Derived a novel formula for generating operators

Connected generating operators with infinite-dimensional representations

Enhanced understanding of differential operators on manifolds

Abstract

Motivated by the classical ideas of generating functions for orthogonal polynomials, we initiate a new line of investigation on "generating operators" for a family of differential operators between two manifolds. We prove a novel formula of the generating operators for the Rankin-Cohen brackets by using higher-dimensional residue calculus. Various results on the generating operators are also explored from the perspective of infinite-dimensional representation theory.