I have a question concerning the ordering of the index in the product symbol. Please take a look at the highlighted indices in the following image:
In equation (4), I read the summation as "starting" at $k = 1$, and "ending" at $k = t$.
In equation (5) however, I am reading the product as "starting" at $i = t$, and "ending" at $i = k+1$.
Have I read this correctly? Or do I have it backwards? Is this what the notation is meant to convey here?
For the sum the order of the finitely many summands doesn't matter, so you need not think about whether the sum starts or ends at $1$.
For the product it won't matter either if the factors are numbers. If they are matrices (operators) then it might. If it does, I think conveying that information in the product sign is not a particularly good idea.
Note that one of the index lists includes $k=1$; the other doesn't. Presumably that matters.