Special values of hyperlogarithms and linear difference schemes
An analytic property of higher logarithms is investigated from the view point of generating functions. It is shown that the logarithmic iterated integrals in the sense of K. T. Chen \[ \int^{1}_{0}(d \log x)^{r_ 1}(d \log (1-x))^{r_ 2}(d \log x)^{r_ 3}..., \] where \(r_ 1\geq 1,r_ 2\geq 1,r_ 3\geq 1,...,r_ n\geq 1\), can be described by the Taylor expansions of an analytic function \(F^*_ n(z_ 1,z_ 2,z_ 3,...)\) at the origin by \[ \frac{1}{r_ 1!r_ 2!r_ 3!...}\frac{\partial^{r_ 1+r_ 2+r_ 3+...}}{\partial z_ 1^{r_ 1} \partial z_ 2^{r_ 2} \partial z_ 3^{r_ 3}...}F^*_ n(0,0,0,...). \] \(F^*_ n(z_ 1,z_ 2,z_ 3,...)\) satisfies holonomic linear difference equations with respect to the displacement \(z_ j\Rightarrow z_ j\pm 1\) over the rational function fields. They are characterized by these equations plus their asymptotic properties. In particular, we have \[ \int^{1}_{0}(d \log x)^{r_ 1}(d \log (1-x))^{r_ 2}= \] \[ \frac{(-1)^{r_ 1}}{r_ 1!r_ 2!}\frac{\partial^{r_ 1+r_ 2}}{\partial z_ 1^{r_ 1} \partial z_ 2^{r_ 2}}[\frac{\Gamma (1+z_ 1)\Gamma (1+z_ 2)}{\Gamma (1+z_ 1+z_ 2)}]_{z_ 1=z_ 2=0}. \] \(F^*_ 3(z_ 1,z_ 2,z_ 3)\) is expressed by the Goursat-Thomae-Whipple hypergeometric function of 3rd order. In generl \(F^*_ n(z_ 1,z_ 2,z_ 3,...)\) have simple integral representations of hypergeometric type.
scientific article; zbMATH DE number 4120540