Thus, this polynomial is closely related to the heptagon. Similarly,
1 - 6 x + 8 x^{3} is closely related to the nonagon, as a factor
of U_{8}(x). In general, the following is true
... but I don't see at Mathworld, or in my references.
Please visit the Mathworld link (Chebyshev
Polynomial of the Second Kind) to see more representations. A
useful recurrence relationship: U_{0}(x) = 1. U_{1}(x)
= 2x. U_{k}_{+2}(x) = 2 x
U_{k}_{+1}(x) - U_{k}(x).
This is easily proved by induction if you use the matrix representation,
but would likely be tricky using the product of cosines relationship.