Identities among some combinatorial objects involving special values of multiple zeta functions

In the article, we establish some identities involving special values of multiple zeta functions among the counting functions of number of representations of an integer by a linear combination of figurate numbers such as triangular numbers, square numbers, pentagonal numbers, etc. More precisely, we provide our result for δₖ(n), rₖ(n) and Nₖᵃ(n) (for a fixed a ≥ 3), the number of representations of n as a sum of k-triangular numbers, as a sum of k-square numbers and as a sum of k-higher figurate numbers (for a fixed a ≥ 3), respectively. Moreover, these identities also occur when one of δₖ(n), rₖ(n) and Nₖᵃ(n) is replaced by the k-colored partition functions.