  It is shown that for any positive integer k and positive parameter lambda
less than 2/k(k+1), the Poisson distribution of order k with parameter lambda
has a unique mode, 0. In addition, the Poisson distribution of order 2 has a
unique mode, 0, if lambda is less than -1 plus the square root of 3. It has two
modes, 0 and 2, if lambda is equal to -1 plus the square root of 3, and it has
a unique mode, 2, if lambda is greater than -1 plus the square root of 3 and
less than 1.
