Remark 5.6. Symmetric sums of type II(p, p, (−1)p−1) share the property that the sums for n odd are zero, because of the change of sign when reversing the direction of summation.

Definition 5.7. A sum of type II(p, p,±x) is called balanced if we can write it in canonical form (5.8) with aj−bj = 2a, j = 1, . . . , p for some constant a. Remark 5.8. In the case of balanced sums we must let a1 = n and b2 = n such that we get b1 = n − 2a and a2 = n + 2a, while the rest of the arguments can be written as aj = cj + a and bj = cj − a. This means that the canonical form of a balanced sum becomes

S(c, n) = n∑

( n

k

) [n + 2a]k[n− 2a]n−k

[cj + a]k[cj − a]n−k xk. (5.10)

Definition 5.9. A sum is called well-balanced if it is balanced and can be written in canonical form (5.10) for some constant a, and such that for some j > 2 we further have aj + bj = n− 2.