This question is related to notation of infinite product.
We know that, $$ \prod_{i=1}^{\infty}x_{i}=x_{1}x_{2}x_{3}\cdots $$
How do I denote $$ \cdots x_{3}x_{2}x_{1} ? $$
One approach could be $$ \prod_{i=\infty}^{1}x_{i}=\cdots x_{3}x_{2}x_{1} $$
I need to use this expression in a bigger expression so I need a good notation for this. Thank you in advance for your help.
Is there any reason to avoid the obvious $\;\; \displaystyle\prod_{i=-\infty}^{-1} x_{-i} \;\;$ ?
(as opposed to dropping the negative signs, like in the approach you suggested)
Sometimes in Clifford algebra when they do products backwards they talk of the "reverse" of the product. I've seen this denoted various ways with tidles: $\widetilde{abc}=cba$ or $(abc)^{\sim}=cba$. If you like them you could consider $$\widetilde{\Pi_{i=1}^\infty a_i}$$ or $$(\Pi_{i=1}^\infty a_i)^\sim$$
If they’re matrices, you can of course simply use $$\left(\prod_{n\ge 0}x_n^T\right)^T\;.$$
In the theory of non-autonomous abstract evolution equations, it is quite costumary to use the followiong notation:
For a family of operators $U_0,U_1,\ldots,U_{n-1}\in\mathcal{L}(X)$, we denote the "time-ordered" product of these operators by \begin{equation*} \prod_{p=0}^{n-1}U_p:=U_{n-1} U_{n-2} \cdots U_1 U_0\quad\mbox{and}\quad\prod_{p=n-1}^{0}U_p:=U_0U_1\cdots U_{n-2} U_{n-1} . \end{equation*}
See Pazy, Page 130.
(With tongue in cheek:) what about this? $$\left(x_n\prod_{i=1}^\infty \right)\;$$