Some binomial formulas for non-commuting operators

2018/01/14 by Peter Kuchment, Kuchment, Peter, Sergey Lvin +1
Mathematics · #16B99 #20F12 #33B9 #44A12 #47B47 #81S05 #92C55 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary 11B65 #Quantum Algebra (math.QA) #Secondary #math.CA #math.QA #msc:11B65 #msc:16B99 #msc:20F12 #msc:33B9 #msc:44A12 #msc:47B47 #msc:81S05 #msc:92C55

paper · pdf · doi:10.48550/arxiv.1801.05308

12 pages

arxiv created 2018/01/14 · arxiv updated 2018/01/17

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 binomial-type 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^λx or by sin λx are basic examples, for which some of these relations appeared unexpectedly as byproducts of an authors' previous medical imaging research.

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