Fix a positive integer m, and let wm = e 2i m be the principal mth root of unity. For each integer j, define the

Fix a positive integer m, and let wm = e 2πi m be the principal mth root of unity. For each integer j, define the segmental function mαj (x)

of x to be the finite Fourier transform mαj (x) = 1 m

m

−1 k=0 exwk mw−jk m .

These functions generalize the hyperbolic trig functions cosh(x) and sinh(x). Prove the following assertions:

(a) mαj (x) = mαj+m(x).

(b) mαj (x + y) = m−1 k=0 mαk(x)mαj−k(y).

(c) mαj (x) = ∞
k=0 xj+km (j+km)! for 0 ≤ j ≤ m − 1.

(d) d dx mαj (x)

= mαj−1(x) .

(e) Consider the differential equation dm dxm f(x) = kf(x) with initial conditions dj dxj f(0) = cj for 0 ≤ j ≤ m − 1, where k and the cj are constants. Show that f(x) = m −1 j=0 cjk− j m mαj (k 1 m x).

(f) The differential equation dm dxm f(x) = kf(x) + g(x) with initial conditions dj dxj f(0) = cj for 0 ≤ j ≤ m − 1 has solution f(x) =  x 0 k− m−1 m mαm−1[k 1 m (x − y)]g(y)dy +
m −1 j=0 cjk− j m mαj (k 1 m x).
(g) limx→∞ e−x mαj (x) = 1 m .
(h) In a Poisson process of intensity 1, e−x mαj (x) is the probability that the number of random points on [0, x] equals j modulo m.
(i) Relative to this Poisson process, let Nx count every mth random point on [0, x]. Then Nx has probability generating function P(s) = e−x m −1 j=0 s− j m mαj (s 1 m x).
(j) Furthermore, Nx has mean E(Nx) = x m − e−x m m −1 j=0 jmαj (x).
(k) limx→∞ 
E(Nx) − x m 
= − m−1 2m .