Multiple zeta values are the real numbers
\begin{equation}
\zeta({\bf a})= \sum_{n_1>\cdots>n_r>0}n_1^{-a_1}\cdots n_r^{-a_r},
\end{equation}
where ${\bf a}=(a_1, \ldots ,a_r) $ is an *admissible composition*,
i.e. a finite sequence of positive integers, with $a_1 \geqslant 2$ when
$r\neq 0$.

The *multiple Apéry-like sums* defined by
\begin{equation}
\sigma({\bf a})=\sum_{n_1>\cdots>n_r>0}\left({2 n_1 \atop
n_1}\right)^{-1}n_1^{-a_1}\cdots n_r^{-a_r}
\end{equation}
when ${\bf a}\neq\varnothing$ and by $\sigma(\varnothing)=1$. We show
that for any admissible composition ${\bf a}$, there exists a finite
formal $\bf Z$-linear combination $\sum \lambda_{\bf b} {\bf b}$ of
admissible compositions such that
\begin{equation}
\zeta({\bf a})=\sum \lambda_{\bf b}\, \sigma({\bf b}).
\end{equation}
The simplest instance of this fact is the identity
\begin{equation}
\sum_{n=1}^{\infty}\frac{1}{n^2}=3\sum_{n=1}^{\infty}\frac{1}{\left({2n
\atop n}\right)n^2}
\end{equation}
discovered by Euler, which expresses that $\zeta(2)=3\,\sigma(2)$. Note
that multiple Apéry-like sums have the advantage on multiple zeta
values to be exponentially quickly convergent.

This allows us to put in a new theoretical context several identities
scattered in the literature, as well as to discover many new interesting
ones. We give new integral formulas for multiple zeta values and
Apéry-like sums. They enable us to give a short direct proof of
Zagier’s formulas for $\zeta(2,\ldots,2,3,2,\ldots,2)$ (D. Zagier,
*Evaluation of the multiple zeta values*
$\zeta(2,\ldots,2,3,2,\ldots,2)$, Annals of Math. **175** (2012),
977–1000) as well as of similar ones in the context of
Apéry-like sums.

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Last updated: 02 Dec 2023