Euler sums of multiple hyperharmonic numbers

Abstract

For k ≔ (k₁, …, k_r) ∈ ℕ^r and n, m ∈ ℕ, we extend the definition of classical hyperharmonic numbers to define the multiple hyperharmonic numbers ${\zeta}_n^{(m)}(k)$ and the Euler sums of multiple hyperharmonic numbers ζ^(m)(q; k)(m + 2 − k₁ ≤ q ∈ ℕ). When k = (k) ∈ ℕ, these sums were first studied by Mezö and Dil around 2010, Dil and Boyadzhiev (2015), and more recently, by Dil, Mezö, and Cenkci, Can, Kargin, Dil, and Soylu, and Li. We show that the multiple hyperharmonic numbers ${\zeta}_n^{(m)}(k)$ can be expressed in terms combinations of products of polynomial in n of degree at most m − 1 and classical multiple harmonic sums with depth ≤ r, and prove that the Euler sums of multiple hyperharmonic numbers ζ^(m) (q; k) can be evaluated by classical multiple zeta values with weight ≤ q + |k| and depth ≤ r + 1.