Ap\'ery-Like Sums and Colored Multiple Zeta Values

TL;DR

This paper surveys recent advances in Apéry-like sums, which generalize Apéry's sums related to zeta values, highlighting their connections to colored multiple zeta values and applications in physics.

Contribution

It provides a comprehensive overview of methods for evaluating Apéry-like sums and proves some conjectural identities, advancing understanding of their structure and relations.

Findings

Connections between Apéry-like sums and colored multiple zeta values

New identities related to Apéry-like sums and zeta values

Methods for computing and relating these sums to Feynman diagram calculations

Abstract

In this article we shall survey some recent progress on the study of Ap\'ery-like sums which are multiple variable generalizations of the two sums Ap\'ery used in his famous proof of the irrationality of $\zeta(2)$ and $\zeta(3)$. We only allow the central binomial coefficients to appear in these infinite sums but they can appear either on the numerator or on the denominator. Special values of both types are closely related to the colored multiple zeta values and have played important roles in the calculations of the $\eps$-expansion of multiloop Feynman diagrams. We will summarize several different approaches to computing these sums and prove a few conjectural identities of Z.-W. Sun as corollaries along the way.