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

DefinitionPartial sums of Schlesinger–Zudilin qMZVs#✓fully proved

For an index \((k_1, \dots, k_r)\) of natural numbers and a natural number \(M\), the partial sum is the formal power series in \(q\) over \(\mathbb{Q}\)

\[\sum_{M > m_1 > \dots > m_r > 0} \prod_{j=1}^{r} \frac{q^{m_j k_j}}{(1 - q^{m_j})^{k_j}},\]

where the sum runs over integers \(m_1, \dots, m_r\) and \(1/(1 - q^m)\) is the inverse in \(\mathbb{Q}[[q]]\). For the empty index it equals \(1\) for every \(M\).

public noncomputable def szQMZVPartial : List ℕ → ℕ → ℚ⟦X⟧
  | [], _ => 1
  | k :: ks, M => ∑ m ∈ Finset.Ioo 0 M, (X ^ m * (1 - X ^ m)⁻¹) ^ k * szQMZVPartial ks m
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Definition of Schlesinger–Zudilin qMZVs (defining sum truncated at m₁ < M).
Used by 5
Used byThe coefficient of \(q\) in \(\zeta_q^{\mathrm{SZ}}(1)\) · Nonempty Schlesinger–Zudilin qMZVs have no constant term · Schlesinger–Zudilin qMZVs · The empty Schlesinger–Zudilin qMZV · Recursion for partial sums of Schlesinger–Zudilin qMZVs

DefinitionSchlesinger–Zudilin qMZVs#✓fully proved

For an index \(k = (k_1, \dots, k_r)\) of natural numbers, the Schlesinger–Zudilin qMZV \(\zeta_q^{SZ}(k)\) is the formal power series in \(q\) over \(\mathbb{Q}\) whose coefficient of \(q^N\), for each \(N \ge 0\), is the coefficient of \(q^N\) in the partial sum

\[\sum_{N \ge m_1 > \dots > m_r > 0} \prod_{j=1}^{r} \frac{q^{m_j k_j}}{(1 - q^{m_j})^{k_j}}\]

over integers \(m_1, \dots, m_r\). In particular \(\zeta_q^{SZ}(\emptyset) = 1\).

When the index is empty or \(k_1 \ge 1\), the summands with \(m_1 > N\) have no term of degree at most \(N\), so \(\zeta_q^{SZ}(k)\) is the full series \(\sum_{m_1 > \dots > m_r > 0} \prod_{j} q^{m_j k_j} / (1 - q^{m_j})^{k_j}\). The paper does not define the value when \(k_1 = 0\).

public noncomputable def szQMZV (k : List ℕ) : ℚ⟦X⟧ :=
  PowerSeries.mk fun N => coeff N (szQMZVPartial k (N + 1))
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Definition of Schlesinger–Zudilin qMZVs (extended version of Ebrahimi-Fard, Manchon and Singer, allowing kⱼ = 0 for j ≥ 2).
Uses 1 · Used by 16
Statement usesPartial sums of Schlesinger–Zudilin qMZVs
Used byThe coefficient of \(q\) in \(\zeta_q^{\mathrm{SZ}}(1)\) · Coefficients of Schlesinger–Zudilin qMZVs as sums of binomial products · Nonempty Schlesinger–Zudilin qMZVs have no constant term · Partitions from Schlesinger–Zudilin qMZVs · The empty Schlesinger–Zudilin qMZV · Schlesinger–Zudilin duality · Spanning set of Schlesinger–Zudilin qMZVs · Stuffle product formula for Schlesinger–Zudilin qMZVs · Shuffle product formula for Schlesinger–Zudilin qMZVs (Singer) · Schlesinger–Zudilin qMZVs of words · SZ-qMZV of the word of a one-entry index · Coefficients of the SZ generating series · Generating series of Schlesinger–Zudilin qMZVs · OOZ-qMZVs via SZ-qMZVs · SZ-qMZVs in the TBZ model · TBZ-qMZVs in the SZ model

TheoremThe empty Schlesinger–Zudilin qMZV#✓fully proved

The Schlesinger–Zudilin qMZV of the empty index is \(1\): \(\zeta_q^{SZ}(\emptyset) = 1\).

public theorem szQMZV_nil : szQMZV [] = 1
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Definition of Schlesinger–Zudilin qMZVs.
Uses 2
Statement usesSchlesinger–Zudilin qMZVs
Proof usesPartial sums of Schlesinger–Zudilin qMZVs

TheoremRecursion for partial sums of Schlesinger–Zudilin qMZVs#✓fully proved

For natural numbers \(k_1\) and \(M\) and an index \(\kappa = (k_2, \dots, k_r)\) (possibly empty), write \(Z_M(k_1, \kappa)\) and \(Z_m(\kappa)\) for the partial sums of Schlesinger–Zudilin qMZVs with bounds \(M\) and \(m\). Then

\[Z_M(k_1, \kappa) = \sum_{0 < m < M} \left(\frac{q^m}{1 - q^m}\right)^{k_1} Z_m(\kappa).\]
public theorem szQMZVPartial_cons (k : ℕ) (ks : List ℕ) (M : ℕ) :
    szQMZVPartial (k :: ks) M =
      ∑ m ∈ Finset.Ioo 0 M, (X ^ m * (1 - X ^ m)⁻¹) ^ k * szQMZVPartial ks m
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Definition of Schlesinger–Zudilin qMZVs.
Uses 1
Statement usesPartial sums of Schlesinger–Zudilin qMZVs

TheoremNonempty Schlesinger–Zudilin qMZVs have no constant term#✓fully proved

For every nonempty index \((k_1, \dots, k_r)\) of natural numbers, the constant term of the Schlesinger–Zudilin qMZV \(\zeta_q^{\mathrm{SZ}}(k_1, \dots, k_r)\) is \(0\).

public theorem coeff_zero_szQMZV_cons (k : ℕ) (ks : List ℕ) :
    coeff 0 (szQMZV (k :: ks)) = 0
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Definition of Schlesinger–Zudilin qMZVs.
Uses 2
Statement usesSchlesinger–Zudilin qMZVs
Proof usesPartial sums of Schlesinger–Zudilin qMZVs

TheoremThe coefficient of \(q\) in \(\zeta_q^{\mathrm{SZ}}(1)\)#✓fully proved

The coefficient of \(q\) in the Schlesinger–Zudilin qMZV \(\zeta_q^{\mathrm{SZ}}(1) = \sum_{m > 0} q^m / (1 - q^m)\) is \(1\).

public theorem coeff_one_szQMZV_singleton_one : coeff 1 (szQMZV [1]) = 1
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Definition of Schlesinger–Zudilin qMZVs.
Uses 2
Statement usesSchlesinger–Zudilin qMZVs
Proof usesPartial sums of Schlesinger–Zudilin qMZVs

DefinitionBlock indices for Schlesinger–Zudilin qMZVs#✓fully proved

For a list of pairs of natural numbers \((k_1, d_1), \dots, (k_r, d_r)\), the block index is the index \((k_1, \{0\}^{d_1 - 1}, \dots, k_r, \{0\}^{d_r - 1})\) obtained by concatenating, for \(j = 1, \dots, r\), the entry \(k_j\) followed by \(d_j - 1\) zeros (no zeros when \(d_j = 0\)).

public def szBlockIndex (kd : List (ℕ × ℕ)) : List ℕ :=
  (kd.map fun p => p.1 :: List.replicate (p.2 - 1) 0).flatten
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Theorem 4 and Theorem 5 (SZ duality), index notation.
Used by 7
Used byCoefficients of Schlesinger–Zudilin qMZVs as sums of binomial products · Recursion for block indices · Block index of the empty list · Schlesinger–Zudilin duality · Coefficients of the SZ generating series · Generating series of Schlesinger–Zudilin qMZVs · Generating series of OOZ-qMZVs, coefficientwise

TheoremBlock index of the empty list#✓fully proved

The block index of the empty list of pairs is the empty index.

public theorem szBlockIndex_nil : szBlockIndex [] = []
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Theorem 5, index notation.
Uses 1
Statement usesBlock indices for Schlesinger–Zudilin qMZVs

TheoremRecursion for block indices#✓fully proved

For natural numbers \(k\), \(d\) and a list \(L\) of pairs of natural numbers, the block index of the list with first pair \((k, d)\) followed by \(L\) is \(k\), followed by \(d - 1\) zeros (none when \(d = 0\)), followed by the block index of \(L\).

public theorem szBlockIndex_cons (k d : ℕ) (kd : List (ℕ × ℕ)) :
    szBlockIndex ((k, d) :: kd) = k :: (List.replicate (d - 1) 0 ++ szBlockIndex kd)
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Theorem 5, index notation.
Uses 1
Statement usesBlock indices for Schlesinger–Zudilin qMZVs

DefinitionPartial Ohno–Okuda–Zudilin sums#✓fully proved

For an index \((k_1, \dots, k_r)\) of natural numbers and a natural number \(M\), the partial OOZ sum is the formal power series in \(q\) over \(\mathbb{Q}\)

\[\sum_{M > m_1 > \dots > m_r > 0} \prod_{j=1}^{r} \frac{1}{(1 - q^{m_j})^{k_j}},\]

the defining sum of \(\zeta_q^{\mathrm{OOZ}}(k_1, \dots, k_r)\) with the factor \(q^{m_1}\) removed and truncated to \(m_1 < M\). For the empty index it is \(1\), for every \(M\) (including \(M = 0\)).

public noncomputable def oozQMZVPartial : List ℕ → ℕ → ℚ⟦X⟧
  | [], _ => 1
  | k :: ks, M => ∑ m ∈ Finset.Ioo 0 M, ((1 - X ^ m)⁻¹) ^ k * oozQMZVPartial ks m
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Definition of OOZ-qMZVs (inner sums of the defining series).
Used by 2
Used byNonempty Ohno–Okuda–Zudilin qMZVs have no constant term · Ohno–Okuda–Zudilin q-multiple zeta values

DefinitionOhno–Okuda–Zudilin q-multiple zeta values#✓fully proved

For an index \((k_1, \dots, k_r)\) of natural numbers with \(r \ge 1\), the (extended) Ohno–Okuda–Zudilin qMZV is the formal power series in \(q\) over \(\mathbb{Q}\)

\[\zeta_q^{\mathrm{OOZ}}(k_1, \dots, k_r) = \sum_{m_1 > \dots > m_r > 0} \frac{q^{m_1}}{(1 - q^{m_1})^{k_1} \cdots (1 - q^{m_r})^{k_r}},\]

and \(\zeta_q^{\mathrm{OOZ}}(\varnothing) = 1\) for the empty index. The summands with \(m_1 > N\) have \(q\)-adic order greater than \(N\), so the coefficient of \(q^N\) is computed from the finitely many summands with \(m_1 \le N\).

public noncomputable def oozQMZV : List ℕ → ℚ⟦X⟧
  | [] => 1
  | k :: ks => PowerSeries.mk fun N =>
      coeff N (∑ m ∈ Finset.Ioo 0 (N + 1), X ^ m * ((1 - X ^ m)⁻¹) ^ k * oozQMZVPartial ks m)
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Definition of OOZ-qMZVs (extended version, k₁ ≥ 1, kᵢ ≥ 0).
Uses 1 · Used by 11
Statement usesPartial Ohno–Okuda–Zudilin sums
Used byClass numbers of finite general linear groups via OOZ-qMZVs · Nonempty Ohno–Okuda–Zudilin qMZVs have no constant term · The OOZ-qMZV of the empty index · Spanning sets of Ohno–Okuda–Zudilin qMZVs · Generating series of OOZ-qMZVs, coefficientwise · OOZ qMZVs respect the OOZ quasi-shuffle product · OOZ-qMZVs of admissible indices via BZ-qMZVs · OOZ-qMZVs via SZ-qMZVs · Duality for restricted OOZ-qMZVs · Duality for OOZ-qMZVs · OOZ-qMZVs are SZ-star qMZVs of the dual index

TheoremThe OOZ-qMZV of the empty index#✓fully proved

The Ohno–Okuda–Zudilin qMZV of the empty index is \(1\): \(\zeta_q^{\mathrm{OOZ}}(\varnothing) = 1\).

public theorem oozQMZV_nil : oozQMZV [] = 1
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Definition of OOZ-qMZVs.
Uses 1
Statement usesOhno–Okuda–Zudilin q-multiple zeta values

TheoremNonempty Ohno–Okuda–Zudilin qMZVs have no constant term#✓fully proved

For every nonempty index \((k_1, \dots, k_r)\) of natural numbers, the constant term of the Ohno–Okuda–Zudilin qMZV \(\zeta_q^{\mathrm{OOZ}}(k_1, \dots, k_r)\) is \(0\).

public theorem coeff_zero_oozQMZV_cons (k : ℕ) (ks : List ℕ) :
    coeff 0 (oozQMZV (k :: ks)) = 0
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Definition of OOZ-qMZVs.
Uses 2
Statement usesOhno–Okuda–Zudilin q-multiple zeta values
Proof usesPartial Ohno–Okuda–Zudilin sums

TheoremCoefficients of Schlesinger–Zudilin qMZVs as sums of binomial products#●ready to prove

Let \(r \ge 0\), and let \(k_1, \dots, k_r\) and \(d_1, \dots, d_r\) be positive integers. For every natural number \(N\), the coefficient of \(q^N\) in the Schlesinger–Zudilin qMZV \(\zeta_q^{\mathrm{SZ}}(k_1, \{0\}^{d_1 - 1}, \dots, k_r, \{0\}^{d_r - 1})\), whose index lists each \(k_j\) followed by \(d_j - 1\) zeros, equals

\[\sum \prod_{j=1}^{r} \binom{n_j - 1}{k_j - 1} \binom{m_j - m_{j+1} - 1}{d_j - 1},\]

where the sum runs over all integers \(m_1 > \dots > m_r > 0\) and \(n_1, \dots, n_r \ge 1\) with \(m_1 n_1 + \dots + m_r n_r = N\), and where \(m_{r+1} = 0\).

These are the coefficients of \(X_1^{k_1 - 1} \cdots X_r^{k_r - 1} Y_1^{d_1 - 1} \cdots Y_r^{d_r - 1} q^N\) on both sides of the paper's identity expressing the generating series \(s(X; Y)\) of these qMZVs as \(\sum \prod_j (1 + X_j)^{n_j - 1} (1 + Y_j)^{m_j - m_{j+1} - 1} q^{m_j n_j}\).

public theorem_wanted coeff_szQMZV_szBlockIndex_eq_sum_choose (r : ℕ) (k d : Fin r → ℕ)
    (hk : ∀ j, 1 ≤ k j) (hd : ∀ j, 1 ≤ d j) (N : ℕ) :
    coeff N (szQMZV (szBlockIndex (List.ofFn fun j => (k j, d j)))) =
      ∑ mn ∈ (Fintype.piFinset fun _ : Fin r => Finset.range (N + 1)) ×ˢ
          (Fintype.piFinset fun _ : Fin r => Finset.range (N + 1)) with
          StrictAnti mn.1 ∧ (∀ j, 0 < mn.1 j) ∧ (∀ j, 0 < mn.2 j) ∧
            ∑ j, mn.1 j * mn.2 j = N,
        ∏ j : Fin r, ((((mn.2 j - 1).choose (k j - 1)) *
          ((mn.1 j - (if h : j.val + 1 < r then mn.1 ⟨j.val + 1, h⟩ else 0) - 1).choose
            (d j - 1)) : ℕ) : ℚ)
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Theorem 4.
Uses 2
Statement usesBlock indices for Schlesinger–Zudilin qMZVs · Schlesinger–Zudilin qMZVs

TheoremSchlesinger–Zudilin duality#●ready to prove

Let \(\zeta_q^{SZ}\) denote the Schlesinger–Zudilin qMZV, let \((k_1, d_1), \dots, (k_r, d_r)\) be pairs of natural numbers with \(k_j \ge 1\) and \(d_j \ge 1\) for all \(j\), and let \(\{0\}^n\) denote \(n\) zeros. Then

\[\zeta_q^{SZ}(k_1, \{0\}^{d_1 - 1}, \dots, k_r, \{0\}^{d_r - 1}) = \zeta_q^{SZ}(d_r, \{0\}^{k_r - 1}, \dots, d_1, \{0\}^{k_1 - 1}).\]

That is, \(\zeta_q^{SZ}\) takes the same value at the block index of the list of pairs and at the block index of the reversed list with each pair swapped.

public theorem_wanted szQMZV_szBlockIndex_dual (kd : List (ℕ × ℕ))
    (hkd : ∀ p ∈ kd, 1 ≤ p.1 ∧ 1 ≤ p.2) :
    szQMZV (szBlockIndex kd) = szQMZV (szBlockIndex (kd.reverse.map Prod.swap))
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Theorem 5 (SZ duality, Zhao).
Uses 2
Statement usesBlock indices for Schlesinger–Zudilin qMZVs · Schlesinger–Zudilin qMZVs

TheoremPartitions from Schlesinger–Zudilin qMZVs#●ready to prove

Let \(\zeta_q^{SZ}\) denote the Schlesinger–Zudilin qMZV, \(\{1\}^r\) the index of \(r\) ones and \(p(N)\) the number of partitions of \(N\). For all natural numbers \(N \ge 1\) and \(R \ge N\), the coefficient of \(q^N\) in \(\sum_{r=1}^{R} \zeta_q^{SZ}(\{1\}^r)\) equals \(p(N)\).

Since the summands with \(r > N\) have no \(q^N\) term, this is the coefficientwise form of \(\sum_{r \ge 1} \zeta_q^{SZ}(\{1\}^r) = \sum_{N \ge 1} p(N) q^N\).

public theorem_wanted sum_coeff_szQMZV_replicate_one_eq_card_partition (N R : ℕ)
    (hN : 1 ≤ N) (hR : N ≤ R) :
    ∑ r ∈ Finset.Icc 1 R, coeff N (szQMZV (List.replicate r 1)) =
      Fintype.card N.Partition
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Lemma 3.
Uses 1
Statement usesSchlesinger–Zudilin qMZVs

TheoremCycle type classifies conjugacy classes of permutations#●ready to prove

For every natural number \(n\), there is a bijection from the set of conjugacy classes of the symmetric group \(S_n\) to the set of partitions of \(n\) that sends the class of each permutation \(\sigma\) to the cycle type of \(\sigma\): the partition of \(n\) formed by the lengths of the cycles of \(\sigma\), fixed points counting as cycles of length \(1\).

public theorem_wanted exists_conjClasses_perm_equiv_partition (n : ℕ) :
    ∃ e : ConjClasses (Equiv.Perm (Fin n)) ≃ (Fintype.card (Fin n)).Partition,
      ∀ σ : Equiv.Perm (Fin n), e (ConjClasses.mk σ) = σ.partition
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Lemma 4.

TheoremNumber of conjugacy classes of the symmetric group#●ready to prove

For every natural number \(n\), the number of conjugacy classes of the symmetric group \(S_n\) equals the number \(p(n)\) of partitions of \(n\).

public theorem_wanted card_conjClasses_perm_eq_card_partition (n : ℕ) :
    Nat.card (ConjClasses (Equiv.Perm (Fin n))) = Fintype.card n.Partition
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Lemma 4.

TheoremClass numbers of finite general linear groups via OOZ-qMZVs#●ready to prove

Let \(K\) be a finite field with \(c\) elements and, for \(n \ge 0\), let \(a_{n,K}\) be the number of conjugacy classes of \(\mathrm{GL}_n(K)\), so that \(a_{0,K} = 1\) because \(\mathrm{GL}_0(K)\) is the trivial group. Write \(\{1\}^r\) for the index consisting of \(r\) ones and \(\zeta_q^{\mathrm{OOZ}}\) for the Ohno–Okuda–Zudilin qMZV. Then for all natural numbers \(N \le R\), \(a_{N,K}\) equals the coefficient of \(q^N\) in \(\sum_{r=0}^{R} (c - 1)^r \zeta_q^{\mathrm{OOZ}}(\{1\}^r)\).

Since the summands with \(r > N\) have no \(q^N\) term, this is the identity of formal power series \(\sum_{n \ge 0} a_{n,K} q^n = \sum_{r \ge 0} (c - 1)^r \zeta_q^{\mathrm{OOZ}}(\{1\}^r)\).

public theorem_wanted card_conjClasses_GL_eq_coeff_sum_oozQMZV (K : Type*) [Field K]
    [Fintype K] (N R : ℕ) (hR : N ≤ R) :
    (Nat.card (ConjClasses (GL (Fin N) K)) : ℚ) =
      ∑ r ∈ Finset.range (R + 1),
        ((Fintype.card K : ℚ) - 1) ^ r * coeff N (oozQMZV (List.replicate r 1))
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Theorem 20.
Uses 1
Statement usesOhno–Okuda–Zudilin q-multiple zeta values

DefinitionPartial sums of general qMZVs#✓fully proved

Let \((k_1, Q_1), \dots, (k_r, Q_r)\) be pairs of natural numbers \(k_j\) and polynomials \(Q_j \in \mathbb{Q}[X]\), and let \(M\) be a natural number. The partial sum is the formal power series in \(q\) over \(\mathbb{Q}\)

\[\sum_{M > m_1 > \dots > m_r > 0} \prod_{j=1}^{r} \frac{Q_j(q^{m_j})}{(1 - q^{m_j})^{k_j}},\]

where the sum runs over integers \(m_1, \dots, m_r\) and \(1/(1 - q^m)\) is the inverse in \(\mathbb{Q}[[q]]\). For \(r = 0\) it equals \(1\) for every \(M\).

public noncomputable def qMZVPartial : List (ℕ × Polynomial ℚ) → ℕ → ℚ⟦X⟧
  | [], _ => 1
  | (k, Q) :: kQ, M => ∑ m ∈ Finset.Ioo 0 M,
      Polynomial.aeval (X ^ m : ℚ⟦X⟧) Q * ((1 - X ^ m)⁻¹) ^ k * qMZVPartial kQ m
Sources
  • Benjamin Brindle, A Unified Approach to qMZVs, INTEGERS 24 (2024), Article A6, Definition of qMZVs (after Bachmann–Kühn), defining sum truncated at m₁ < M.
Used by 2
Used byNonempty general qMZVs have no constant term · General q-multiple zeta values
14 fully proved6 ready to prove

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