noncommutative power series

Noncommutative power series extend the idea of infinite algebraic sums, like regular power series, to variables where order matters—meaning that switching the order of multiplication changes the result. Unlike standard variables that commute, noncommutative variables represent entities such as matrices or elements in certain algebraic structures where \(AB \neq BA\). These series are infinite sums of products of such variables, allowing mathematicians to analyze and work with complex systems where order influences outcomes—crucial in areas like quantum mechanics, control theory, and operator algebras.