Old and new powerful tools for the normal ordering problem and noncommutative binomials

TL;DR

This paper develops general formulas for noncommutative exponentiation and binomial expansions, applying them to cases with specific commutator structures, and introduces a new operator connecting different ordering schemes.

Contribution

It provides new formal formulas for noncommutative exponentials and binomials, including cases with quadratic and monomial commutators, and introduces a novel operator linking normal and antinormal orderings.

Findings

Derived formulas for noncommutative exponential functions.

Extended binomial theorem for specific commutator types.

Introduced a new operator connecting ordering schemes.

Abstract

In this paper, we derive formal general formulas for noncommutative exponentiation and the exponential function, while also revisiting an unrecognized, and yet powerful theorem. These tools are subsequently applied to derive counterparts for the exponential identity $e^{A+B} = e^A e^B$ and the binomial theorem $(A+B)^n = \sum \binom{n}{k} A^k B^{n-k}$ when the commutator $[B, A]$ is either an arbitrary quadratic polynomial or a monomial in $A$ or $B$. Analogous formulas are found when the commutator is bivariate. Furthermore, we introduce a novel operator bridging between the normal and antinormal ordered forms.