Let $A,B$ be square matrices. I am interested in an accepted notation for expressing the terms in the sum when $(A+B)^k$ for some integer k is expanded.
For example, when $k=2$, $(A+B)^2 = A^2 + AB + BA + B^2$. I'd like to express this as an iterated sum where the outer sum refers to the number of times the factor $A$ appears in each addend. For example, I'm looking for an existing notation such that $(A+B)^2 = \sum_{i=0}^2 \sigma(A,B,i)$ where $\sigma(A,B,0) = A^2, \sigma(A,B,1) = AB + BA, \sigma(A,B,2) = B^2$ or along these lines.
Of course, $\sigma(A,B,i)$ contains another summation as can be seen in $\sigma(A,B,1)$. I am dealing with an arbitrary $k$ which is why I'm looking for an established notation if one exists. I also do not assume that $A,B$ commute.
I'm not sure if the obtained formula in this paper is of common use or if it exactly satisfies your needs, but it gives a closed formula for $(A + B)^n$ where $A$ and $B$ are elements in a non-necessarily commutative algebra with identity (e.g. a matrix algebra over a field); the closed formula is given making use of commutators and also regarding each element as a linear transformation on the chosen algebra via left multiplication.