Parametric Ap\'{e}ry-type Series and Hurwitz-type Multiple Zeta Values

TL;DR

This paper generalizes Apéry-type series by incorporating parameters and explores their connections to Hurwitz-type multiple zeta values using iterated integrals, extending previous results in the field.

Contribution

It introduces parametric Apéry-type series involving binomial coefficients and Hurwitz-type sums, establishing explicit relations with Hurwitz-type multiple zeta values.

Findings

Derived explicit formulas linking parametric Apéry-type series to Hurwitz-type multiple zeta values.

Extended previous results to include parametric binomial coefficients and Hurwitz-type sums.

Utilized iterated integrals to establish these relations.

Abstract

In this paper, we extend the main results of a 2024 \emph{Advances in Applied Mathematics} paper \cite{XuZhao2021c} about Ap\'{e}ry-type series involving central binomial coefficients and the multiple ($t-$)harmonic sums to parametric Ap\'{e}ry-type series involving parametric binomial coefficients and Hurwitz-type multiple harmonic (star) sums. In particular, we will establish many explicit relations between parametric Ap\'{e}ry-type series involving one or two parametric binomial coefficients and Hurwitz-type multiple zeta values (with $r$-variables) by using the method of iterated integrals.