The Combinatorics of Harmonic Sums and Polylogarithms at negative integer multi- indices

Let Y0 = {ys}s≥0 be an infinite alphabet. We define Y ∗0 to be the (free) monoid of words on the alphabet Y0. Then each of elements w ∈ Y ∗0 can be writen in the form w = ys1... ysr for any r-uplet (s1,..., sr) ∈ Nr. Let r ∈ N, and z ∈ C such that ∣z ∣< 1, then the following Polylogarithm is well defined Li−s1,...,sr(z) ∶ = ∑ n1>...>nr>0 z n1ns11... n sr r. (1) The Taylor expansion of the function (1 − z)−1 Lis1,...,sr(z) is given by Li−s1,...,sr(z) 1 − z = ∑N≥0H−s1,...,sr(N) zN, where the coefficient H−s1,...,sr ∶ N Ð→ Q is an arithmetic function, also called Harmonic sum, which can be expressed as follows H−s1,...,sr(N) ∶ = ∑ N≥n1>...>nr>0n s1 1... n sr r. (2) Then it can be checked that H−w(N) ∈ C[N] and Li−w(z) ∈ C[z] for any w ∈ Y ∗0. Moreover, the sets {H−w(N)}w∈Y