Show that the multinomial distribution with index m and probability vector 1, ,k has cumulant generating function

Show that the multinomial distribution with index m and probability vector π1, …,πk has cumulant generating function m {k (θ + ξ) − k (θ)}, where k(θ) = log[Σexp(θj)] and

πi = exp (θi)/∑exp (θj).

Hence show that the first four cumulants are where, for example, δijk = 1 if i = j = k and zero otherwise, and no summation is implied where indices are repeated at the same level.