Special values of hyperlogarithms and linear difference schemes
type: Item
instance of: scholarly article
title: Special values of hyperlogarithms and linear difference schemes
zbMATH Open document ID: 0684.33010
author: Kazuhiko Aomoto
published in: Illinois Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 33E15
Mathematics Subject Classification ID: 32S40
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 4120540
zbMATH Keywords: hyperlogarithms
zbMATH Keywords: iterated integrals
zbMATH Keywords: holonomic linear difference equations
MaRDI profile type: Publication
Recommended article: Special values of generalized polylogarithms
Recommended article: Special values of hypergeometric functions over finite fields
Recommended article: Special values of multiple polylogarithms
Recommended article: Special values of the hypergeometric series
Recommended article: Special Values of the Hypergeometric Series
Recommended article: Q4381445
Recommended article: Special values of fractional hypergeometric functions for function fields
Recommended article: Special values of Hilbert modular functions
Recommended article: Special values of hypergeometric functions and periods of CM elliptic curves
description: scientific article; zbMATH DE number 4120540