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The shuffle relation of fractions from multiple zeta values. (English) Zbl 1222.11108

Multiple zeta values (MZVs) are special values of the multivariable complex functions \[ \zeta(s_1,\ldots,s_k)=\sum_{n_1>\ldots n_s>0}\frac{1}{n_1^{s_1}\ldots n_k^{s_k}} \] for positive integers \(s_i\), \(1 \leq i \leq k\), and \(s_1\geq 2\). If, for variables \(u_i\), \(n_i=u_i+\ldots+u_k\), \(1 \leq i \leq k\), the MZVs can be represented as rational fraction \[ \zeta(s_1,\ldots,s_k)=\sum_{u_1,\ldots,u_k \geq 1}\frac{1}{(u_1+\dots+u_k)^{s_1}(u_2+\dots+u_k)^{s_2}\ldots u_k^{s_k}}. \]
The authors obtain an explicit product formula for any two MZV fractions using the general double shuffle framework, and show that such fractions have cannonical integral representations.

MSC:

11M41 Other Dirichlet series and zeta functions
11M99 Zeta and \(L\)-functions: analytic theory
40B05 Multiple sequences and series
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