I have the following function $$ B(n)=x \alpha ^n-y \alpha ^n q^n + \epsilon \left(x \alpha ^{n+1}-y \alpha ^{n+1} q^{n+1}\right). $$ Here, I want to find the product $B(n)^2$ or $B(n)*B(n+1)$ where the variables $x$ and $y$ are non-commutative i.e. $xy\neq yx$. Is there any way to achieve this in Mathematica ?
For example, when I try to find the product in Mathematica, I find that
$$ B(n)^2 = -2 x y \epsilon ^2 \alpha ^{2 n+2} q^{n+1}-2 x y \epsilon \alpha ^{2 n+1} q^{n+1}+y^2 \epsilon ^2 \alpha ^{2 n+2} q^{2 n+2}+2 y^2 \epsilon \alpha ^{2 n+1} q^{2 n+1}+y^2 \alpha ^{2 n} q^{2 n}-2 x y \epsilon \alpha ^{2 n+1} q^n-2 x y \alpha ^{2 n} q^n+x^2 \epsilon ^2 \alpha ^{2 n+2}+2 x^2 \epsilon \alpha ^{2 n+1}+x^2 \alpha ^{2 n} $$ and $$ B(n)*B(n+1) = -x y \epsilon ^2 \alpha ^{2 n+3} q^{n+1}-x y \epsilon ^2 \alpha ^{2 n+3} q^{n+2}-2 x y \epsilon \alpha ^{2 n+2} q^{n+1}-x y \epsilon \alpha ^{2 n+2} q^{n+2}-x y \alpha ^{2 n+1} q^{n+1}+y^2 \epsilon ^2 \alpha ^{2 n+3} q^{2 n+3}+2 y^2 \epsilon \alpha ^{2 n+2} q^{2 n+2}+y^2 \alpha ^{2 n+1} q^{2 n+1}-x y \epsilon \alpha ^{2 n+2} q^n-x y \alpha ^{2 n+1} q^n+x^2 \epsilon ^2 \alpha ^{2 n+3}+2 x^2 \epsilon \alpha ^{2 n+2}+x^2 \alpha ^{2 n+1}. $$ But it must be that $xy \neq yx$.
NonCommutativeMultiply
? $\endgroup$B[n]**B[n+1]
? $\endgroup$