A restricted sum formula among multiple zeta values
For positive integers \(\alpha_1,\alpha_2,\dots,\alpha_r\) with \(\alpha_r\geq 2\), the multiple zeta value or \(r\)-fold Euler sum is defined as \[ \zeta(\alpha):= \zeta(\alpha_1,\dots, \alpha_r)= \sum_{1\leq k_1<\cdots< k_r} k^{-\alpha_1}_1\cdots k^{-\alpha_r}_r. \] There is a celebrated sum formula among multiple zeta values as \[ \sum_{|\alpha|= m}\zeta(\alpha_1,\alpha_2,\dots, \alpha_r+ 1)= \zeta(m+1), \] where \(\alpha_1,\dots, \alpha_r\) range over all positive integers with \(|\alpha|= \alpha_1+\cdots+ \alpha_r= m\) in the summation. In the present paper, the authors prove that for all positive integers \(m\) and \(q\) with \(m\geq q\), and a nonnegative integer \(p\), \[ \sum_{|\alpha|= m}\zeta(\{1\}^p,\alpha_1,\alpha_2,\dots, \alpha_q+1)= \sum_{|c|= p+q} \zeta(c_1,c_2, \dots, c_{p+1}+(m-q)+1). \] Here \(\{1\}^p\) denotes \(p\) repetitions of \(1\). When \(p=0\) and \(q=r\), this is precisely the sum formula. Such a formula can be used to determine explicitly some multiple zeta values of lower weights more precisely than the sum formula.
scientific article; zbMATH DE number 5542340
scientific article