Fourier series of sums of products of r -derangement functions

A derangement is a permutation that has no fixed point and the derangement number d m is the number of fixed point-free permutations on an m element set. A generalization of the derangement numbers are the r -derangement numbers and their natural companions are the r -derangement polynomials. In this paper we will study three types of sums of products of r -derangement functions and find Fourier series expansions of them. In addition, we will express them in terms of Bernoulli functions. As immediate corollaries to this, we will be able to express the corresponding three types of polynomials as linear combinations of Bernoulli polynomials.