=+11. For a fixed positive integer n, we define the generalized hyperbolic functions [146] nj (x) of

=+11. For a fixed positive integer n, we define the generalized hyperbolic functions [146] nαj (x) of x as the finite Fourier transform coefficients nαj (x) = 1 n

n

−1 k=0 exuk n u−jk n , where un = e2πi/n is the nth principal root of unity. These functions generalize the hyperbolic trigonometric functions cosh(x) and sinh(x). Prove the following assertions:

(a) nαj (x) = nαj+n(x).

(b) nαj (x + y) = n−1 k=0 nαk(x)nαj−k(y).

(c) nαj (x) = ∞

k=0 xj+kn

(j+kn)! for 0 ≤ j ≤ n − 1.

(d) d dx

nαj (x)



= nαj−1(x) .

(e) limx→∞ e−xnαj (x) = 1 n .

(f) In a Poisson process of intensity 1, e−x nαj (x) is the probability that the number of random points on [0, x] equals j modulo n.

(g) Relative to this Poisson process, let Nx count every nth random point on [0, x]. Then Nx has probability generating function P(s) = e−x n

−1 j=0 s− j n nαj (s 1

n x).

146 6. Poisson Processes

(h) Furthermore, Nx has mean E(Nx) = x n − e−x n

n

−1 j=0 jnαj (x).

(i) limx→∞ 

E(Nx) − x n



= −n−1 2n .