Some binomial formulas for noncommuting operators
Abstract
Let $D$ and $U$ be linear operators in a vector space (or more generally, elements of an associative algebra with a unit). We establish binomialtype identities for $D$ and $U$ assuming that either their commutator $[D,U]$ or the second commutator $[D,[D,U]]$ is proportional to $U$. Operators $D=d/dx$ (differentiation) and $U$ multiplication by $e^{\lambda x}$ or by $\sin \lambda x$ are basic examples, for which some of these relations appeared unexpectedly as byproducts of an authors' previous medical imaging research.
 Publication:

arXiv eprints
 Pub Date:
 January 2018
 DOI:
 10.48550/arXiv.1801.05308
 arXiv:
 arXiv:1801.05308
 Bibcode:
 2018arXiv180105308K
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Quantum Algebra;
 Primary 11B65;
 16B99;
 20F12;
 47B47;
 Secondary;
 33B9;
 44A12;
 92C55;
 81S05
 EPrint:
 12 pages