• 1 Institute of Mathematical Sciences [Chennai] [IMSc]

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 $\delta_k(n)$, $r_{k}(n)$ and $\mathcal{N}_{k}^{a}(n)$ (for a fixed $a \ge 3$), 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 $a \ge 3$), respectively. Moreover, these identities also occur when one of $\delta_k(n)$, $r_{k}(n)$ and $\mathcal{N}_{k}^{a}(n)$ is replaced by the $k$-colored partition functions.

• 05A10 - Factorials, binomial coefficients, combinatorial functions
• 05A15 - Exact enumeration problems, generating functions
• 05A19 - Combinatorial identities, bijective combinatorics
• 05A30 - \(q\)-calculus and related topics
• 11M32 - Multiple Dirichlet series and zeta functions and multizeta values
• 33B10 - Exponential and trigonometric functions