On multiple zeta values of extremal height (Q2800088)

In this article, the authors establish three identities involving multiple zeta values.NEWLINENEWLINEFor multiple zeta values, there are two types of definition: multiple zeta values (MZV) defined by sums with strict inequalities in the summation NEWLINE\[NEWLINE \zeta (k_1,\,\dots ,\,k_r)=\sum\limits_{0<m_1<\dots <m_r}\frac{1}{{m_1}^{k_1}\dots {m_r}^{k_r}}, NEWLINE\]NEWLINE and defined by non-strict inequalities (MZSV) NEWLINE\[NEWLINE \zeta ^*(k_1,\,\dots ,\,k_r)=\sum\limits_{0<m_1\leq \dots \leq m_r}\frac{1}{{m_1}^{k_1}\dots {m_r}^{k_r}}. NEWLINE\]NEWLINE Let \(\mathbf{k} = (k_1,\dots, k_r),\,k_i\in \mathbb{N},\,k_r > 1\), and \(\mathbf{a}+\mathbf{b}\) means the vector \(\mathbf{a}+\mathbf{b}=(a_1+b_1,\dots,a_r+b_r)\). Next, \(w(\mathbf{k}) := k_1 +\dots +k_r, \,d(\mathbf{k}) := r, \, h(\mathbf{k}) := \sharp\{i| k_i > 1, 1 \leq i \leq r\}\) are called, respectively, the weight, the depth, and the height of the index set \(\mathbf{k}\).NEWLINENEWLINEFor any integers \(r,\,k\geq 1\) the following explicit formula for the height-one NEWLINE\[NEWLINE \zeta (\underbrace{1,\dots 1}_{{r-1}},k+1)=\sum\limits_{j=1}^{\min(r,\,k)}(-1)^{j-1} \sum\limits_{w(\mathbf{a})=k, w(\mathbf{b})=r\atop{d(\mathbf{a}=d(\mathbf{b}=j))}}\zeta(\mathbf{a}+\mathbf{b}) NEWLINE\]NEWLINE is obtained and also a Shuffle-regularized sum formula NEWLINE\[NEWLINE \sum\limits_{w(\mathbf{k})=r+k\atop{d(\mathbf{k})=r}}\zeta^{III}(\mathbf{k})= (-1)^{r-1}\zeta (\underbrace{1,\dots 1}_{{r-1}},k+1) NEWLINE\]NEWLINE is obtained.NEWLINENEWLINEFinally, the authors give a kind of sum formula for the maximal-height MZVs in the form of generating function. More precisely they establishe that if \(T(k)\) is the sum of all multiple zeta values of weight \(k\) and of maximal height NEWLINE\[NEWLINE T(k):=\sum\limits_{k_1+\dots + k_r=k\atop{r\geq 1,\,\forall k_i\geq 2}}\zeta (k_1,\,\dots ,\,k_r) NEWLINE\]NEWLINE then NEWLINE\[NEWLINE 1+\sum\limits_{k+2}^{\infty}T(k)x^k= \bigg(1+\sum\limits_{n=1}^{\infty}\zeta ^*(2,\,\dots ,\,2)x^{2n} \bigg) \bigg(1+\sum\limits_{n=1}^{\infty}\zeta (3,\,\dots ,\,3)x^{3n} \bigg). NEWLINE\]