Alternating multiple zeta values, and explicit formulas of some Euler-Apery-type series

TL;DR

This paper derives explicit formulas for Euler-Apery-type series involving binomial coefficients and harmonic numbers using iterated integrals and alternating multiple zeta values, revealing their connections to fundamental constants.

Contribution

It introduces new explicit formulas for certain Euler-Apery-type series and demonstrates their reducibility to well-known constants using advanced integral techniques.

Findings

Explicit formulas for Euler-Apery-type series involving zeta values

Reduction of series to ln(2), zeta, and alternating zeta values

Numerical evaluations of numerous special series

Abstract

In this paper, we study some Euler-Ap\'ery-type series which involve central binomial coefficients and (generalized) harmonic numbers. In particular, we establish elegant explicit formulas of some series by iterated integrals and alternating multiple zeta values. Based on these formulas, we further show that some other series are reducible to ln(2), zeta values, and alternating multiple zeta values by considering the contour integrals related to gamma functions, polygamma functions and trigonometric functions. The evaluations of a large number of special Euler-Ap\'ery-type series are presented as examples.