[Paper Review] Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series

Kam Cheong Au|arXiv (Cornell University)|Jul 8, 2020

Advanced Mathematical Identities10 citations

TL;DR

This paper presents a systematic method for evaluating one-dimensional integrals involving polylogarithms and generalized polylogarithms, using iterated integrals and regularization techniques. It derives closed-form evaluations for numerous Apéry-type infinite series and definite integrals, expressing results in terms of colored multiple zeta values (CMZVs), multiple zeta values, and special constants like Catalan's constant and the Dirichlet beta function.

ABSTRACT

We give systematic method to evaluate a large class of one-dimensional integral relating to multiple zeta values (MZV) and colored MZV. We also apply the technique of iterated integrals and regularization to elucidate the nature of some infinite series involving binomial coefficients. This technique can be applied to many Ap\\'ery-type infinite sums.

Motivation & Objective

• To develop a general method for evaluating one-dimensional integrals involving ordinary and generalized polylogarithms.
• To systematically evaluate infinite series with binomial coefficients and inverse central binomial coefficients, particularly Apéry-type sums.
• To express results in terms of colored multiple zeta values (CMZVs), especially at levels 2 and 4, and to clarify their algebraic structure.
• To provide explicit evaluations of previously conjectured or difficult infinite series, including those involving $\operatorname{Li}_n(1/2)$, $\operatorname{Li}_n((1+i)/2)$, and $\beta(4)$.
• To establish a framework for reducing complex polylogarithmic integrals and series into algebraically structured constants using iterated integrals and regularization.

Proposed method

• Utilizes iterated integrals on paths in the complex plane, defined via pullbacks of differential forms, to represent integrands involving polylogarithms.
• Applies regularization techniques to handle divergent or singular integrals arising in the evaluation of generalized polylogarithmic functions.
• Employs the Hoffman-Racinet algebraic framework to manipulate and reduce iterated integrals into combinations of standard MZV and CMZV bases.
• Uses the $S_3 \times S_4$ symmetry group action on $4$-admissible rational functions to generate equivalent integral representations and simplify evaluation.
• Applies Fourier-Legendre expansions and functional equations to relate integrals to known special values of $\operatorname{Li}_n(z)$ and $\zeta(s)$.
• Leverages a custom Mathematica package (available on ResearchGate) to compute and verify explicit integrals involving $\operatorname{Li}_n(x)$, particularly for $n \leq 8$.

Experimental results

Research questions

• RQ1 How can one-dimensional integrals involving polylogarithms be systematically evaluated using iterated integrals and regularization?
• RQ2 What is the algebraic structure of infinite series involving $\binom{2n}{n}^{-1}$, $\binom{3n}{n}^{-1}$, and harmonic sums, and how can they be reduced to known constants?
• RQ3 Which combinations of $\operatorname{Li}_n(1/2)$, $\operatorname{Li}_n((1+i)/2)$, $\beta(4)$, and $\zeta(3)$ appear in closed forms of Apéry-type series?
• RQ4 How do level 2 and level 4 colored multiple zeta values ($\textsf{CMZV}^2_n$, $\textsf{CMZV}^4_n$) arise naturally in the evaluation of such integrals and series?
• RQ5 Can the evaluation of integrals involving $\operatorname{Li}_2(-4x/(1-x)^2)$ and $\log^k(1+x)$ be reduced to known special constants using algebraic and analytic techniques?

Key findings

• The series $\sum_{n=2}^{\infty} \frac{H_{n-1}^{(2)}}{n^3} \left[2^n \binom{2n}{n}^{-1}\right]$ evaluates to $\frac{\pi^3 C}{24} - \pi \beta(4) - \frac{3\pi^2 \zeta(3)}{128} + \frac{527 \zeta(5)}{256} + \frac{1}{384} \pi^4 \log 2$, with $C$ the Catalan constant.
• The series $\sum_{n=1}^{\infty} \frac{1}{n^5 2^n \binom{3n}{n}}$ is evaluated in closed form as $4\pi \Im\left(\operatorname{Li}_4\left(\frac{1}{2} + \frac{i}{2}\right)\right) + 3\pi \beta(4) - \frac{51}{2} \operatorname{Li}_5\left(\frac{1}{2}\right) - 15 \operatorname{Li}_4\left(\frac{1}{2}\right) \log 2 + \frac{\pi^2 \zeta(3)}{4} + \frac{9\zeta(5)}{2} - 3\zeta(3) \log^2 2 - \frac{97}{240} \log^5 2 + \frac{41}{144} \pi^2 \log^3 2 - \frac{61}{960} \pi^4 \log 2$, confirming a conjecture by Borwein.
• The integral $\int_0^1 \frac{\operatorname{Li}_2(-4x/(1-x)^2) \operatorname{Li}_3(1-x^2)}{x} dx$ evaluates to a combination of $\operatorname{Li}_4(1/2)$, $\zeta(3)^2$, $\zeta(5)\log 2$, $\pi^6$, $\log^4 2$, and $\pi^4 \log^2 2$.
• The integral $\int_0^1 \frac{\log^2(1-x) \log^2 x \log^3(1+x)}{x} dx$ yields a highly complex expression involving $\operatorname{Li}_5(1/2)^2$, $\operatorname{Li}_6(1/2)$, $\zeta(3)^2$, $\pi^8$, and various log-powers, with no missing level 2 weight 8 CMZV term.
• The sum $\sum_{n=1}^{\infty} \frac{H_n^{s_1,\dots,s_k}}{n^s} a_n^{\pm 1}$ lies in $\textsf{CMZV}^2_S$, $\sum_{n=1}^{\infty} \frac{H_n^{s_1,\dots,s_k}}{n^s} a_n^{-2}$ in $\textsf{CMZV}^4_S$, and $\sum_{n=1}^{\infty} \frac{H_n^{s_1,\dots,s_k}}{n^s} a_n^2$ in $\frac{1}{\pi} \textsf{CMZV}^4_{S+1}$, where $a_n = 4^{-n} \binom{2n}{n}$.
• A complete reduction of level 4 weight 5 CMZVs is computed, and a Mathematica package is provided for symbolic evaluation of such integrals and series.