Duality formulas of the Special Values of Multiple Polylogarithms

A triple integral analog of a multiple zeta value

We establish the triple integral evaluation \[ \int_{1}^{\infty} \int_{0}^{1} \int_{0}^{1} \frac{dz \, dy \, dx}{x(x+y)(x+y+z)} = \frac{5}{24} ζ(3), \] as well as the equivalent polylogarithmic double sum \[ \sum_{k=1}^{\infty} \sum_{j=k}^{\infty} \frac{(-1)^{k-1}}{k^{2}} \, \frac{1}{j \, 2^{j}} = \frac{13}{24} ζ(3). \] This double sum is related to, but less approachable than, similar sums studied by Ramanujan. It is also reminiscent of Euler's formula $ζ(2,1) = ζ(3)$, which is the simplest instance of duality of multiple polylogarithms. We review this duality and apply it to derive a compani…

Special Values of Multiple Polylogarithms

Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove se…

Special Values of Generalized Log-sine Integrals

We study generalized log-sine integrals at special values. At $π$ and multiples thereof explicit evaluations are obtained in terms of Nielsen polylogarithms at $\pm1$. For general arguments we present algorithmic evaluations involving Nielsen polylogarithms at related arguments. In particular, we consider log-sine integrals at $π/3$ which evaluate in terms of polylogarithms at the sixth root of unity. An implementation of our results for the computer algebra systems Mathematica and SAGE is provided.

Special values of trigonometric Dirichlet series and Eichler integrals

We provide a general theorem for evaluating trigonometric Dirichlet series of the form $\sum_{n \geq 1} \frac{f (πn τ)}{n^s}$, where $f$ is an arbitrary product of the elementary trigonometric functions, $τ$ a real quadratic irrationality and $s$ an integer of the appropriate parity. This unifies a number of evaluations considered by many authors, including Lerch, Ramanujan and Berndt. Our approach is based on relating the series to combinations of derivatives of Eichler integrals and polylogarithms.

特殊函数中对偶性的简短综述

A short survey of duality in special functions

本文以教程形式综述特殊函数（尤其是（$q$-)Askey框架中的正交多项式）的对偶性质，内容源自作者2017年R.P. Agarwal纪念讲座的第一部分。

Duality transformation formulas for multiple elliptic hypergeometric series of type $BC$

New duality transformation formulas are proposed for multiple elliptic hypergeometric series of type $BC$ and of type $C$. Various transformation and summation formulas are derived as special cases to recover some previously known results.