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.

arXiv comment16 pages, RevTex; v2: add a footnote on Fock-Segal-Bargmann transform (post publication)

References in corpus (12)

• Umbral Calculus, Difference Equations and the Discrete Schroedinger Equation
• Erlangen Programme at Large: An Overview
• Heisenberg Algebra, Umbral Calculus and Orthogonal Polynomials
• (Discrete) Almansi Type Decompositions: An umbral calculus framework based on $\mathfrak{osp}(1|2)$ symmetries
• Operator Covariant Transform and Local Principle
• Erlangen Programme at Large 3.1: Hypercomplex Representations of the Heisenberg Group and Mechanics
• The Real and Complex Techniques in Harmonic Analysis from the Point of View of Covariant Transform
• Cross-Toeplitz Operators on the Fock--Segal--Bargmann Spaces and Two-Sided Convolutions on the Heisenberg Group
• Tokens: An Algebraic Construction Common in Combinatorics, Analysis, and Physics
• 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