Hi,
I do not know how to do that in an elegant way, but I noted that this
Product[i, {i, Complement[Range[1, 4], {3}]}]
is a way to exempt certain indices (in this case 3) from the product. In oder to evaluate the products we need to fix i. In my example Manipulate does that for us.
Manipulate[
Product[Subscript[x, j] - Subscript[x, i], {j,
Complement[Range[1, 3], {i}]}]/
Product[Subscript[x, i] - Subscript[x, k], {k,
Complement[Range[1, 3], {i}]}], {i, 1, 3, 1}]
You can also get the results for different
$i$ in form of a table:
Table[Product[
Subscript[x, j] - Subscript[x, i], {j,
Complement[Range[1, 3], {i}]}]/
Product[Subscript[x, i] - Subscript[x, k], {k,
Complement[Range[1, 3], {i}]}], {i, 1, 3}]
which results in:
I am sure that there is a super-easy way of doing this, but in principle the functions above should do the trick.
Cheers,
Marco