Apéry-Like Sums and Colored Multiple Zeta Values

Abstract

In this article, we survey recent progress in the study of Apéry-like sums, which are multivariable generalizations of the two sums Apéry used in his famous proof of the irrationality of $\zeta (2)$ and $\zeta (3)$. We focus exclusively on infinite sums where central binomial coefficients appear in either the numerator or the denominator. Special values of both types are closely related to colored multiple zeta values and play important roles in calculating the $\epsilon$-expansion of multiloop Feynman diagrams. Finally, we summarize several distinct approaches to computing these sums.

30 pages