chapter 8

9 Transformations of Sums of Type II(2,2,z)

Pages 8

This theorem tells us that with each possible value of z, there are up to five other formulas easily obtainable, giving the six values

z, 1/z, 1− z, 1/(1− z), z/(z − 1), 1− 1/z.

The cases having fewer than six different possibilities are (a) z = 1, giving at most 1− z = 0; (b) z = 1/z = −1, related to 1− z = 1− 1/z = 2 and 1/(1 − z) = z/(z − 1) = 12 , giving a group of 3; and (c) the sixth root of unity, z = 1/(1 − z) = ei π3 = 1 − 1/z = 12 + i

2 , related to 1/z = 1 − z = z/(z − 1) = e−i π3 = 12 − i

2 , giving a group of 2 complex conjugates.