Multiple zeta values

Multiple zeta and zeta-star values, Hoffman bases, word algebras, cyclic sum formulas, finite and q-analogue multiple zeta values.

• Multiple-zeta indices
• Multiple-zeta and multiple-zeta-star series
• Reversing multiple-zeta summation conventions
• Faà di Bruno's formula and nested sums of reciprocal squares
• Finite multiple zeta values of level N
• Height one multiple zeta values
• Cyclic sum formula
• Hoffman algebra and its bases
• q-analogues of multiple zeta values

TheoremConvergence of multiple zeta series#✓fully proved

Let \(\mathbf{k} = (k_1, \dots, k_r)\) be an admissible index, that is, a nonempty index of positive integers with \(k_1 \ge 2\). Then the strict multiple-zeta series of \(\mathbf{k}\), the family of terms \((m_1^{k_1} \cdots m_r^{k_r})^{-1}\) indexed by the strictly decreasing tuples \(m_1 > \cdots > m_r\) of positive integers, is summable.

theorem summable_strictSummand (index : Index) (hindex : index.IsAdmissible) :
    Summable (strictSummand index)

Sources

• arXiv:2307.09867, lines 60–63, and arXiv:2309.07569, lines 76–80; equivalent increasing-variable formulations occur in arXiv:2304.08722, 2312.13525, 2608.10675, 2608.15480, and 2608.22280 after reversing both the variables and exponents.

Uses 6 · Used by 1

Statement usesMultiple-zeta index · Admissible index · Depth of an index · Strictly decreasing tuples · Multiple zeta summand

Proof usesEntry of a multiple-zeta index

Used byCyclic sum formula for strict multiple zeta values of indices

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