Transmutations from the Covariant Transform on the Heisenberg Group and an Extended Umbral Principle

Abstract

We discuss several seemingly assorted objects: the umbral calculus, generalised translations and associated transmutations, symbolic calculus of operators. The common framework for them is representations of the Weyl algebra of the Heisenberg group by ladder operators. Transporting various properties between different implementations we review some classic results and new opportunities.

16 pages, RevTex; v2: add a footnote on Fock-Segal-Bargmann transform (post publication)

References in corpus (6)

• Umbral Calculus, Difference Equations and the Discrete Schroedinger Equation
• Heisenberg Algebra, Umbral Calculus and Orthogonal Polynomials
• Cross-Toeplitz Operators on the Fock--Segal--Bargmann Spaces and Two-Sided Convolutions on the Heisenberg Group
• Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration
• Global Functional calculus, lower/upper bounds and evolution equations on manifolds with boundary
• Spectrum as the Support of Functional Calculus