Some variations of multiple zeta values (Q680330)

For positive integers \(r,\alpha_1,\alpha_2,\ldots,\alpha_r\) with \(\alpha_r\geqslant 2\), the convergent infinite series \[ \zeta(\alpha_1,\alpha_2,\ldots,\alpha_r)=\sum_{0<k_1<k_2<\cdots<k_r}k_1^{-\alpha_1}k_2^{-\alpha_2}\cdots k_r^{-\alpha_r} \] defines a multiple zeta value, which has an iterated integral representation \[ \zeta(\alpha_1,\alpha_2,\ldots,\alpha_r)=\int_0^1\frac{dt}{1-t}\left(\frac{dt}{t}\right)^{\alpha_1-1}\cdots\frac{dt}{1-t}\left(\frac{dt}{t}\right)^{\alpha_r-1}, \] where for \(1\)-forms \(\omega_i(t)\), we have \[ \int_0^1\omega_1(t)\omega_2(t)\cdots \omega_l(t)=\int_{0<t_1<t_2<\cdots<t_l<1}\omega_1(t_1)\omega_2(t_2)\cdots \omega_l(t_l). \] In the paper under review, the authors add the factor \(\frac{t^b}{(1-t)^a}\) to the front of the \(1\)-forms in the iterated integral of some multiple zeta values, and the factor \(\frac{(1-t)^c}{t^d}\) to the back of the \(1\)-forms. They provide an infinite series representations of this new integral and its dual, and then apply suitable differential operators to obtain relations among some variations of multiple zeta values. For example, they consider the iterated integral \[ \int_0^1\frac{t^bdt}{(1-t)^{a+p+1}}\left(\frac{dt}{t}\right)^{r-1}\frac{(1-t)^adt}{t^{b+1}} \] and its dual. The authors obtain many interesting relations, especially they show an analogue of the sum formula of the multiple zeta values \[ \sum_{\alpha_1+\alpha_2+\cdots+\alpha_r=m+r\atop \alpha_1,\alpha_2,\ldots,\alpha_r\geqslant 1}\sum_{0<k_1<k_2<\cdots<k_r}k_1^{-\alpha_1}k_2^{-\alpha_2}\cdots k_r^{-\alpha_r}(k_1-1)^{-1}=\zeta(r), \] where \(m\) and \(r\) are nonnegative integers with \(r\geqslant 2\).