q-analogues of multiple zeta values

Partial sums of Schlesinger–Zudilin qMZVs to Partial sums of general qMZVs

• Partial sums of Schlesinger–Zudilin qMZVs
• Schlesinger–Zudilin qMZVs
• The empty Schlesinger–Zudilin qMZV
• Recursion for partial sums of Schlesinger–Zudilin qMZVs
• Nonempty Schlesinger–Zudilin qMZVs have no constant term
• The coefficient of \(q\) in \(\zeta_q^{\mathrm{SZ}}(1)\)
• Block indices for Schlesinger–Zudilin qMZVs
• Block index of the empty list
• Recursion for block indices
• Partial Ohno–Okuda–Zudilin sums
• Ohno–Okuda–Zudilin q-multiple zeta values
• The OOZ-qMZV of the empty index
• Nonempty Ohno–Okuda–Zudilin qMZVs have no constant term
• Coefficients of Schlesinger–Zudilin qMZVs as sums of binomial products
• Schlesinger–Zudilin duality
• Partitions from Schlesinger–Zudilin qMZVs
• Cycle type classifies conjugacy classes of permutations
• Number of conjugacy classes of the symmetric group
• Class numbers of finite general linear groups via OOZ-qMZVs
• Partial sums of general qMZVs

General q-multiple zeta values to Sums over partitions are invariant under transposition

• General q-multiple zeta values
• General qMZV of the empty index
• Nonempty general qMZVs have no constant term
• Spaces of general qMZVs
• The constant series one lies in every space of qMZVs
• Bradley–Zhao q-multiple zeta values
• Bradley–Zhao qMZV of the empty index
• Nonempty Bradley–Zhao qMZVs have no constant term
• Block index of a list of pairs
• Block index of a prepended pair
• Spanning set of Schlesinger–Zudilin qMZVs
• Spanning set of Bradley–Zhao qMZVs
• Spanning sets of Ohno–Okuda–Zudilin qMZVs
• Bradley–Zhao duality for q-multiple zeta values
• q-Ohno relations for Bradley–Zhao qMZVs
• Partitions in Stanley coordinates
• Membership criterion for partitions in Stanley coordinates
• Transposed partition in Stanley coordinates
• Transpose of a one-part partition
• Sums over partitions are invariant under transposition

Coefficients of qMZVs as polynomial functions on partitions to Takeyama–Bradley–Zhao qMZVs

• Coefficients of qMZVs as polynomial functions on partitions
• Polynomial functions on partitions characterize \(\mathcal{Z}_q\)
• Quasi-shuffle product
• The empty word is a left unit for the quasi-shuffle product
• The stuffle product of two letters
• SZ-qshuffle product
• SZ-qshuffle of \(y\) with itself
• Word of an index
• Word of the empty index
• Reading an index off a word in \(p\) and \(y\)
• Reading off a leading block of \(p\)s
• Schlesinger–Zudilin qMZVs of words
• SZ-qMZV of the word of a one-entry index
• Quasi-shuffle relations for general qMZVs
• Stuffle product formula for Schlesinger–Zudilin qMZVs
• Shuffle product formula for Schlesinger–Zudilin qMZVs (Singer)
• Quasi-shuffle relations for Bradley–Zhao qMZVs
• Analytic properties of general qMZVs
• Letters of the Takeyama–Bradley–Zhao model as qMZV data
• Takeyama–Bradley–Zhao qMZVs

TBZ-qMZV of the empty index to Coefficients of the SZ generating series

• TBZ-qMZV of the empty index
• The TBZ model extends the BZ model
• The Takeyama–Bradley–Zhao model spans \(\mathcal{Z}_q\)
• The spaces \(\mathcal{Z}^\circ_{q,d}\)
• The constant \(1\) lies in \(\mathcal{Z}^\circ_{q,d}\)
• \(\mathcal{Z}^\circ_{q,d}\) is contained in \(\mathcal{Z}_{q,d}\)
• Okounkov polynomials
• The Okounkov polynomials \(p_2\) and \(p_3\)
• Okounkov \(q\)-multiple zeta values
• Okounkov qMZV of the empty index
• Okounkov qMZVs span \(\mathcal{Z}^\circ_{q,1}\)
• Eulerian q-series
• Coefficients of the Eulerian q-series
• Partial sums of bi-brackets
• Bi-brackets
• Empty bi-bracket
• Generating series of bi-brackets
• Coefficients of the bi-bracket generating series
• Generating series of Schlesinger–Zudilin qMZVs
• Coefficients of the SZ generating series

The logarithm series \(\ln(1 + T)\) to Weighted chain sums

• The logarithm series \(\ln(1 + T)\)
• Low-order coefficients of the logarithm series
• Bi-bracket generating series as a sum of exponentials
• Partition relation for bi-brackets
• Translation between bi-brackets and the Schlesinger–Zudilin model
• Generating series of OOZ-qMZVs, coefficientwise
• Brackets, Okounkov qMZVs and Bradley–Zhao qMZVs span \(\mathcal{Z}^\circ_{q,1}\)
• Weighted quasi-shuffle product
• The empty word is a left unit for the weighted quasi-shuffle
• Bachmann's coefficients
• Bachmann's coefficient \(\lambda_{1,1}^1\)
• Bachmann's product of bi-bracket letters
• Bi-brackets form a quasi-shuffle algebra
• The map \(T\) of the OOZ quasi-shuffle product
• The OOZ map fixes the empty word
• The OOZ map on a one-letter word
• Stuffle product of formal linear combinations of words
• The empty word is a left unit for the stuffle product
• OOZ qMZVs respect the OOZ quasi-shuffle product
• Weighted chain sums

Marker word of a list of blocks to Duality for OOZ-qMZVs

• Marker word of a list of blocks
• Chain sum of the single block \((1, 1)\)
• Combinatorial form of Bradley–Zhao duality
• The SZ dual index
• The SZ dual of \((2)\)
• Bradley–Zhao dual index
• Bradley–Zhao dual of \((3)\)
• The tail part of the map \(U\)
• The map \(U\) from OOZ to BZ words
• \(U\) fixes the empty index
• The map \(U\) fixes \(z_2\)
• The tail part of the map \(V\)
• The map \(V\) from OOZ to SZ words
• The map \(V\) fixes \(py\)
• OOZ-qMZVs of admissible indices via BZ-qMZVs
• The map \(U\) is a bijection on combinations of admissible indices
• OOZ-qMZVs via SZ-qMZVs
• The map \(V\) is a bijection on combinations of SZ-admissible indices
• Duality for restricted OOZ-qMZVs
• Duality for OOZ-qMZVs

Partial sums of SZ-star qMZVs to Duality for TBZ-qMZVs

• Partial sums of SZ-star qMZVs
• SZ-star qMZVs
• The empty SZ-star qMZV
• The coefficient of \(q\) in \(\zeta_q^{SZ,\star}(1)\)
• OOZ-qMZVs are SZ-star qMZVs of the dual index
• The map from TBZ words to SZ indices
• The TBZ-to-SZ map on the letter \(\bar{1}\)
• The map \(V_{TBZ}\) from SZ indices to TBZ words
• The map \(V_{TBZ}\) on the index \((1)\)
• TBZ-qMZVs in the SZ model
• SZ-qMZVs in the TBZ model
• Duality for TBZ-qMZVs

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