What is the concept of two series commutes with each other?
For example let, $f(x)=\sum a_n x^n$ and $g(x)=\sum b_n y^n$, then how do we commute $f$ ad $g$?
If just like the concept of composition of function, then we have $$ f(x) \circ g(x)=\sum a_n (\sum b_n y)^n=\sum b_n (\sum a_n x)^n=g(x) \circ f(x). $$
Is it the concept of ''commutes of series" ?
But how it is true or can be verified?
Let $f = \sum a_n x^n$ and $g = \sum b_n x^n$ be two formal series. Then $$ f \circ g = \sum a_n g(x)^n = \sum a_n \left(\sum b_k x^k \right)^n $$ Now, since $$ \left(\sum b_k x^k \right)^n = \sum_{i_1 + \dotsm + i_k = n} b_{i_1} \dotsm b_{i_k} x^n $$ one gets $$ f \circ g = \sum a_n \left(\sum_{i_1 + \dotsm + i_k = n} b_{i_1} \dotsm b_{i_k}\right)x^n $$ Therefore $f$ and $g$ commute (for the composition) if and only if, for all $n$, $$ a_n \left(\sum_{i_1 + \dotsm + i_k = n} b_{i_1} \dotsm b_{i_k}\right) = b_n \left(\sum_{i_1 + \dotsm + i_k = n} a_{i_1} \dotsm a_{i_k}\right) $$