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Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series

Kam Cheong Au

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 Apéry-type infinite sums.

Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series

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 Apéry-type infinite sums.

Paper Structure

This paper contains 14 sections, 19 theorems, 109 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Iterated integral
  4. Hoffman-Racinet algebra
  5. Regularization
  6. Some low level, low weight reduction
  7. One dimensional definite integral
  8. Some examples
  9. Admissible rational function
  10. Infinite sums that are reducible to CMZVs
  11. Apéry-like series involving harmonic numbers
  12. Fourier-Legendre expansion of generalized polylogarithm
  13. Series involving binomial coefficient squared and harmonic numbers
  14. Rapidly converging series

Key Result

Proposition 2.1

Iterated integral enjoys the following properties: where $\gamma^{-1}$ is the reverse path of $\gamma$. where $\gamma_2(1) = \gamma_1(0)$, here $\gamma_1\gamma_2$ means composition of two paths, first $\gamma_2$, then $\gamma_1$. the last sum is over certain elements of symmetric group $S_{n+m}$, it can also be viewed as shuffle product between $\omega_1\cdots \omega_n$ and $\omega_{n+1}\cdots

Theorems & Definitions (54)

  • Proposition 2.1
  • Example 2.2: Harmonic number
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Example 2.6: Generalized polylogarithm
  • Theorem 2.7
  • Example 3.1
  • ...and 44 more