functions - Sum or Product with Exclusions

Is there a built-in feature for handling things like:

$$\sum_{\substack{i=0\\i\ne j}}^n\frac{a-a_i}{a_i-a_j}$$ and $$\prod_{\substack{i=0\\i\ne j}}^n\frac{a-a_i}{a_i-a_j}$$

or should I work out some sort of Do statement?

Answer

The documentation for Product[] gives a nice example that you can adapt to your needs:

With[{j = 2, n = 6},
     Sum[(a - Subscript[a, i])/(Subscript[a, i] - Subscript[a, j]),
         {i, Complement[Range[0, n], {j}]}]]

(a - Subscript[a, 0])/(Subscript[a, 0] - Subscript[a, 2]) +
(a - Subscript[a, 1])/(Subscript[a, 1] - Subscript[a, 2]) +
(a - Subscript[a, 3])/(-Subscript[a, 2] + Subscript[a, 3]) +
(a - Subscript[a, 4])/(-Subscript[a, 2] + Subscript[a, 4]) +

(a - Subscript[a, 5])/(-Subscript[a, 2] + Subscript[a, 5]) +
(a - Subscript[a, 6])/(-Subscript[a, 2] + Subscript[a, 6])

and similarly for Product[]. Alternatively, you can do

With[{j = 2, n = 6},
     Sum[(a - Subscript[a, i])/(Subscript[a, i] - Subscript[a, j]),
         {i, DeleteCases[Range[0, n], j]}]]