=+9. Consider a power series f(x) = ∞

m=0 cmxm with radius of convergence r > 0. Prove that

∞

m=k mod n cmxm = 1 n

n

−1 j=0 u−jk n f(uj nx)

for un = e2πi/n and any x with |x| < r. As a special case, verify the identity

∞

m=k mod n

 p m



= 2p n

n

−1 j=0 cos  (p − 2k)jπ

n



cosp

 jπ

n



for any positive integer p.