# Product Notation for Multiplication in Reverse Order

Is there a standard notation for multiplication in reverse order? For example consider the problem

$$x_{k+1} = A_k x_k$$

where $x_i \in \mathbb{R}^n$ and $A_i \in M_n(\mathbb{R})$, ($i=0,1,2,\dots$) without any further assumptions on $A_i$. The solution to this problem is

$$x_k = A_{k-1} \dots A_1 A_0 x_0$$

Obviously, writing

$$x_k = \prod_{i=0}^{k-1} A_i x_0$$

is wrong because of the multiplication order. But I'm also unconfortable to use

$$x_k = \prod_{i=k-1}^{0} A_i x_0$$

as the notation suggests that we need to increment $i$, not decrement it. Also, one may need to write $\prod_{i=k}^{m} A_i$. Then, we need to explicitly state $m < k$ to imply ordering.

I can always write the open form but as similar products frequently occur in calculations in my work I want to express them in a more compact way. The question is, is there a standard and nice notation to do so? There are some suggestions in this question, but I think none of them are "nice enough" to work with.

I came up with the notation

$$x_k = \coprod_{i=k-1}^{0} A_i x_0$$

to imply decrementing $i$ rather than increment it, but I don't know if it is used in somewhere else or there is a standard notation.

$$x_k = \prod_{i=0}^{k-1} A_{k-i-1} x_0$$

(You seem to be fixated on putting things in the bounds of the operator rather than in the operand.)