On a generalization of Chen's iterated integrals (Q1048928)

The author considers generalizations of Chen's iterated integrals. For \(\alpha\) and \(\beta\) are holomorphic 1-forms on a given complex manifold \(M\), \(\gamma\) is a piecewise smooth path in \(M\), \[ \int_\gamma \alpha^{r-1} \beta^{s}:= \frac{(-1)^{s-1}}{\Gamma(r) \Gamma(s)}\int_0^1 \left( \int_0^z \gamma^* \alpha\right)^{r-1} \left( \int_0^z \gamma^* \beta\right)^{s-1}\gamma^* \beta \] for any \(r,s\) \textit{complex} such that the integral in the right converges. For the particular case of \(M=\mathbb{P}^1\setminus \{0,1,\infty\}\), the author defines more general integrals of the form \[ \int_0^1 f_1(z) \frac{dz}{z} \left(\frac{dz}{z}\right)^{s_1-1} \dots\;f_l(z) \frac{dz}{z} \left(\frac{dz}{z}\right)^{s_l-1} \] where \(f_1,\dots,f_k\) are holomorphic functions satisfying certain properties. Such integrals yields certain special \textit{complex} values of multiple zeta values, multiple polylogarithms and multiple Hurwitz zeta functions. Finally, the author considers applications to complex iterated derivatives and a direct topological proof of the monodromy of polylogarithms.