Hyperharmonic Euler sums

DefinitionGeneralized hyperharmonic numbers#✓fully proved

For natural numbers \(n\) and \(k\) and a complex number \(r\), the generalized hyperharmonic number \(H_n^{[r]}(k)\) is \(0\) when \(k > n\); when \(k \le n\) it is the elementary symmetric polynomial of degree \(n - k\) in \(r, r + 1, \ldots, r + n - 1\) divided by \(n!\), that is,

\[H_n^{[r]}(k) = \frac{1}{n!} \sum_{0 \le j_1 < \cdots < j_{n-k} < n} \prod_{i=1}^{n-k} (r + j_i).\]

The source defines \(H_n^{[r]}(k)\) in (2.4) by \((-1)^k k! \, H_n^{[r]}(k) = D_x^k \binom{n+r-x-1}{n}\) at \(x = 0\), where \(D_x^k\) is the \(k\)-th derivative in \(x\); the explicit form above follows from it. For \(r = 1\) it is the depth-\(k\) multiple harmonic number, and for \(r \notin \{0, -1, \ldots, 1 - n\}\) it equals \(\binom{n+r-1}{n} \sum_{1 \le i_1 < \cdots < i_k \le n} \prod_{j=1}^{k} 1/(i_j + r - 1)\), see (2.10).

noncomputable def generalizedHyperharmonic (n k : ℕ) (r : ℂ) : ℂ :=
  if k ≤ n then
    (∑ s ∈ (Finset.range n).powersetCard (n - k), ∏ j ∈ s, (r + j)) / n.factorial
  else 0

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Section 2, (2.4) and (2.10).

Used by 14

Used byGeneralized hyperharmonic numbers vanish for \(k > n\) · The value \(H_1^{(r)}(1) = 1\) · The value \(H_1^{(r)}(0) = r\) · Generalized hyperharmonic numbers with both indices zero · Generalized hyperharmonic Euler sum · Duality for hyperharmonic Euler sums · Hyperharmonic Euler sums as sums over all indices · Hyperharmonic Euler sums with half-integer parameter and shift · Hyperharmonic Euler sums with integer parameter and half-integer shift · Polygamma structure of hyperharmonic Euler sums · Quarter-shift hyperharmonic Euler sums · Products of quarter-shift hyperharmonic Euler sums · Hyperharmonic Euler sums with rational parameters · Expansion of the hyperharmonic zeta function in height one multiple zeta values

TheoremGeneralized hyperharmonic numbers vanish for \(k > n\)#✓fully proved

For all natural numbers \(n\) and \(k\) with \(n < k\) and every complex number \(r\), the generalized hyperharmonic number \(H_n^{[r]}(k)\) equals \(0\).

theorem generalizedHyperharmonic_of_lt {n k : ℕ} (h : n < k) (r : ℂ) :
    generalizedHyperharmonic n k r = 0

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Section 2, (2.4).

Uses 1 · Used by 1

Statement usesGeneralized hyperharmonic numbers

Used byHyperharmonic Euler sums as sums over all indices

TheoremGeneralized hyperharmonic numbers with both indices zero#✓fully proved

For every complex number \(r\), the generalized hyperharmonic number \(H_0^{[r]}(0)\) equals \(1\).

@[simp]
theorem generalizedHyperharmonic_zero_zero (r : ℂ) : generalizedHyperharmonic 0 0 r = 1

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Section 2, (2.4).

Uses 1

Statement usesGeneralized hyperharmonic numbers

TheoremThe value \(H_1^{(r)}(0) = r\)#✓fully proved

For every complex number \(r\), the generalized hyperharmonic number \(H_1^{[r]}(0)\) equals \(r\).

theorem generalizedHyperharmonic_one_zero (r : ℂ) : generalizedHyperharmonic 1 0 r = r

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Section 2, (2.4).

Uses 1

Statement usesGeneralized hyperharmonic numbers

TheoremThe value \(H_1^{(r)}(1) = 1\)#✓fully proved

For every complex number \(r\), the generalized hyperharmonic number \(H_1^{[r]}(1)\) equals \(1\).

theorem generalizedHyperharmonic_one_one (r : ℂ) : generalizedHyperharmonic 1 1 r = 1

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Section 2, (2.4).

Uses 1

Statement usesGeneralized hyperharmonic numbers

DefinitionGeneralized hyperharmonic Euler sum#✓fully proved

For an integer \(k \ge 0\), complex numbers \(r\) and \(a\) and an integer \(p\), the generalized hyperharmonic Euler sum is \(S(k, r; p; a) = \sum_{n \ge k} H_n^{[r]}(k) / (n + a)^p\), where \(H_n^{[r]}(k)\) is the generalized hyperharmonic number. A term whose denominator \((n + a)^p\) vanishes counts as \(0\), and \(S(k, r; p; a) = 0\) when the series is not summable.

noncomputable def hyperharmonicEulerSum (k : ℕ) (r : ℂ) (p : ℤ) (a : ℂ) : ℂ :=
  ∑' n : ℕ, generalizedHyperharmonic (n + k) k r / (((n + k : ℕ) : ℂ) + a) ^ p

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 1 display.

Uses 1 · Used by 9

Statement usesGeneralized hyperharmonic numbers

Used byDuality for hyperharmonic Euler sums · Beta-function evaluation of hyperharmonic Euler sums · Hyperharmonic Euler sums as sums over all indices · Hyperharmonic Euler sums with half-integer parameter and shift · Hyperharmonic Euler sums with integer parameter and half-integer shift · Polygamma structure of hyperharmonic Euler sums · Quarter-shift hyperharmonic Euler sums · Products of quarter-shift hyperharmonic Euler sums · Hyperharmonic Euler sums with rational parameters

TheoremHyperharmonic Euler sums as sums over all indices#✓fully proved

For every integer \(k \ge 0\), complex numbers \(r\) and \(a\), and integer \(p\), write \(H_n^{[r]}(k)\) for the generalized hyperharmonic number. Then the hyperharmonic Euler sum \(S(k, r; p; a) = \sum_{n \ge k} H_n^{[r]}(k) / (n + a)^p\) equals \(\sum_{n \ge 0} H_n^{[r]}(k) / (n + a)^p\), because \(H_n^{[r]}(k) = 0\) for \(n < k\).

Both sides are the sums of the series in the sense of unconditional summation, with the value \(0\) assigned to a series that does not converge, and a term with \(n + a = 0\) and \(p \ne 0\) is read as \(0\), so the identity holds without any hypothesis.

theorem hyperharmonicEulerSum_eq_tsum (k : ℕ) (r : ℂ) (p : ℤ) (a : ℂ) :
    hyperharmonicEulerSum k r p a =
      ∑' n : ℕ, generalizedHyperharmonic n k r / ((n : ℂ) + a) ^ p

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 1 display.

Uses 3

Statement usesGeneralized hyperharmonic numbers · Generalized hyperharmonic Euler sum

Proof usesGeneralized hyperharmonic numbers vanish for \(k > n\)

DefinitionOdd multiple harmonic numbers#✓fully proved

For natural numbers \(k\) and \(n\), the depth-\(k\) odd harmonic number is the real number

\[O_n(k) = \sum_{1 \le i_1 < \cdots < i_k \le n} \prod_{j=1}^{k} \frac{1}{i_j - 1/2}.\]

In particular \(O_n(0) = 1\) (the empty product) and \(O_n(k) = 0\) for \(k > n\) (the empty sum).

noncomputable def oddMultipleHarmonic (k n : ℕ) : ℝ :=
  ∑ s ∈ (Finset.Icc 1 n).powersetCard k, ∏ i ∈ s, ((i : ℝ) - 1 / 2)⁻¹

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, (1.8) and (2.11).

Used by 18

Used byCatalan–odd harmonic sum with denominator \(4^n n\) · Catalan–odd harmonic sums at the shift \(1/2\) · Central binomial sum over \(\binom{n+m}{m} (n + 1/2)^2\) · Odd harmonic central binomial sum over \(\binom{n+m}{m} (n + 1/2)^2\) · Central binomial odd harmonic sum with denominator \(4^n (n + 1)^2\) · Central binomial odd harmonic sums over \(\binom{n+m}{m}\) lie in the period algebra · Central binomial odd harmonic sums over \(\binom{n+m}{m}\) at half-integer shifts · Central binomial odd harmonic sums over \(\binom{n+m}{m}\) at the shift \(1/4\) · Central binomial odd harmonic sums over \(\binom{n+m}{m}\) at the shift \(3/4\) · Products of quarter-shift central binomial odd harmonic sums · Odd harmonic central binomial sums at integer shifts · Odd harmonic central binomial sums with weight \(\binom{n}{m}\) at integer shifts · Odd harmonic central binomial sums with weight \(\binom{n}{m}\) at half-integer shifts · Odd harmonic central binomial sums with weight \(\binom{n}{m}\) at shift \(1/4\) · Odd harmonic central binomial sums with weight \(\binom{n}{m}\) at shift \(3/4\) · Vanishing of odd multiple harmonic numbers of large depth · The first odd harmonic number · Odd multiple harmonic numbers of depth zero

TheoremOdd multiple harmonic numbers of depth zero#✓fully proved

For every natural number \(n\), the depth-zero odd harmonic number \(O_n(0)\) equals \(1\).

@[simp]
theorem oddMultipleHarmonic_zero (n : ℕ) : oddMultipleHarmonic 0 n = 1

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, (2.11).

Uses 1

Statement usesOdd multiple harmonic numbers

TheoremVanishing of odd multiple harmonic numbers of large depth#✓fully proved

For natural numbers \(k\) and \(n\) with \(k > n\), the depth-\(k\) odd harmonic number \(O_n(k)\) equals \(0\).

theorem oddMultipleHarmonic_of_lt {k n : ℕ} (h : n < k) : oddMultipleHarmonic k n = 0

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, (2.11).

Uses 1

Statement usesOdd multiple harmonic numbers

TheoremThe first odd harmonic number#✓fully proved

The depth-one odd harmonic number \(O_1(1)\) equals \(2\).

theorem oddMultipleHarmonic_one_one : oddMultipleHarmonic 1 1 = 2

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, (2.11).

Uses 1

Statement usesOdd multiple harmonic numbers

DefinitionThe algebra generated by \(\log 2\), \(\pi^2\) and odd zeta values#✓fully proved

This is the \(\mathbb{Q}\)-subalgebra of \(\mathbb{C}\) generated by \(\log 2\), \(\pi^2\) and the zeta values \(\zeta(3), \zeta(5), \zeta(7), \dots\) (that is, \(\zeta(2j+1)\) for all integers \(j \ge 1\)). It is written \(\mathbb{Q}[\log 2, \pi^2, \zeta(3), \zeta(5), \dots]\).

noncomputable def eulerSumPeriodAlgebra : Subalgebra ℚ ℂ :=
  Algebra.adjoin ℚ
    ({(Real.log 2 : ℂ), (π : ℂ) ^ 2} ∪ Set.range fun j : ℕ => riemannZeta (2 * j + 3))

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 2.

Used by 9

Used byCatalan–odd harmonic sums at the shift \(1/2\) · Central binomial odd harmonic sums over \(\binom{n+m}{m}\) lie in the period algebra · Central binomial odd harmonic sums over \(\binom{n+m}{m}\) at half-integer shifts · Odd harmonic central binomial sums at integer shifts · Odd harmonic central binomial sums with weight \(\binom{n}{m}\) at integer shifts · Odd harmonic central binomial sums with weight \(\binom{n}{m}\) at half-integer shifts · Generators of the Euler sum period algebra · Hyperharmonic Euler sums with half-integer parameter and shift · Hyperharmonic Euler sums with integer parameter and half-integer shift

TheoremGenerators of the Euler sum period algebra#✓fully proved

The numbers \(\log 2\) and \(\pi^2\) and the odd zeta values \(\zeta(2j + 3)\) for all integers \(j \ge 0\) lie in the \(\mathbb{Q}\)-subalgebra \(\mathbb{Q}[\log 2, \pi^2, \zeta(2j+1)_{j \ge 1}]\) of \(\mathbb{C}\) that they generate.

theorem generators_mem_eulerSumPeriodAlgebra :
    (Real.log 2 : ℂ) ∈ eulerSumPeriodAlgebra ∧ (π : ℂ) ^ 2 ∈ eulerSumPeriodAlgebra ∧
      ∀ j : ℕ, riemannZeta (2 * j + 3) ∈ eulerSumPeriodAlgebra

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 2.

Uses 1

Statement usesThe algebra generated by \(\log 2\), \(\pi^2\) and odd zeta values

DefinitionThe algebra generated by \(\log 2\), \(\pi\), odd zeta values and even beta values#✓fully proved

For integers \(s \ge 2\) let \(\beta(s) = \sum_{n \ge 0} (-1)^n / (2n+1)^s\) be the value of the Dirichlet beta function. This is the \(\mathbb{Q}\)-subalgebra of \(\mathbb{C}\) generated by \(\log 2\), \(\pi\), the zeta values \(\zeta(2j+1)\) for all integers \(j \ge 1\) and the beta values \(\beta(2j)\) for all integers \(j \ge 1\). It is written \(\mathbb{Q}[\log 2, \pi, \zeta(2j+1)_{j \ge 1}, \beta(2j)_{j \ge 1}]\).

noncomputable def eulerSumBetaPeriodAlgebra : Subalgebra ℚ ℂ :=
  Algebra.adjoin ℚ
    ({(Real.log 2 : ℂ), (π : ℂ)} ∪ Set.range (fun j : ℕ => riemannZeta (2 * j + 3)) ∪
      Set.range fun j : ℕ =>
        ((∑' n : ℕ, (-1 : ℝ) ^ n / (2 * n + 1 : ℝ) ^ (2 * j + 2) : ℝ) : ℂ))

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 6.

Used by 8

Used byCentral binomial odd harmonic sums over \(\binom{n+m}{m}\) at the shift \(1/4\) · Central binomial odd harmonic sums over \(\binom{n+m}{m}\) at the shift \(3/4\) · Products of quarter-shift central binomial odd harmonic sums · Odd harmonic central binomial sums with weight \(\binom{n}{m}\) at shift \(1/4\) · Odd harmonic central binomial sums with weight \(\binom{n}{m}\) at shift \(3/4\) · Generators of the Euler sum beta period algebra · Quarter-shift hyperharmonic Euler sums · Products of quarter-shift hyperharmonic Euler sums

TheoremGenerators of the Euler sum beta period algebra#✓fully proved

The numbers \(\log 2\) and \(\pi\), the zeta values \(\zeta(2j + 3)\) for all integers \(j \ge 0\), and the Dirichlet beta values \(\beta(2j + 2) = \sum_{n \ge 0} (-1)^n / (2n + 1)^{2j+2}\) for all integers \(j \ge 0\) lie in the \(\mathbb{Q}\)-subalgebra \(\mathbb{Q}[\log 2, \pi, \zeta(2j+1)_{j \ge 1}, \beta(2j)_{j \ge 1}]\) of \(\mathbb{C}\) that they generate.

theorem generators_mem_eulerSumBetaPeriodAlgebra :
    (Real.log 2 : ℂ) ∈ eulerSumBetaPeriodAlgebra ∧ (π : ℂ) ∈ eulerSumBetaPeriodAlgebra ∧
      (∀ j : ℕ, riemannZeta (2 * j + 3) ∈ eulerSumBetaPeriodAlgebra) ∧
      ∀ j : ℕ, ((∑' n : ℕ, (-1 : ℝ) ^ n / (2 * n + 1 : ℝ) ^ (2 * j + 2) : ℝ) : ℂ) ∈
        eulerSumBetaPeriodAlgebra

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 6.

Uses 1

Statement usesThe algebra generated by \(\log 2\), \(\pi\), odd zeta values and even beta values

TheoremBeta-function evaluation of hyperharmonic Euler sums#●ready to prove

Let \(k, p \ge 0\) be integers and let \(r, a\) be complex numbers with \(\operatorname{Re}(r) < p + 1\) and \(\operatorname{Re}(a) > -k\), such that \(r\) is not a positive integer and \(a\) is not a nonpositive integer. Write \(H_n^{[r]}(k)\) for the generalized hyperharmonic number and let \(S(k, r; p + 1; a) = \sum_{n \ge k} H_n^{[r]}(k) / (n + a)^{p+1}\). Then

\[p! \, k! \, S(k, r; p + 1; a) = (-1)^p D_a^p D_r^k B(a, 1 - r),\]

where \(B(a, 1 - r) = \Gamma(a) \Gamma(1 - r) \cdot (1/\Gamma)(a + 1 - r)\), with \(1/\Gamma\) the entire reciprocal of the Gamma function (zero at the poles of \(\Gamma\)), and \(D_a^p D_r^k\) means the \(k\)-th complex derivative in \(r\) followed by the \(p\)-th complex derivative in \(a\).

The source also states the identity at positive integers \(r\), after removing the singularity of \(\Gamma(1 - r)\), and at nonpositive integers \(a\), and extends it by analytic continuation; those cases are not included here.

public theorem_wanted hyperharmonicEulerSum_eq_iteratedDeriv_beta (k p : ℕ) (r a : ℂ)
    (hr : r.re < p + 1) (ha : -(k : ℝ) < a.re) (hr' : ∀ m : ℕ, r ≠ m + 1)
    (ha' : ∀ m : ℕ, a ≠ -m) :
    (p.factorial * k.factorial : ℂ) * hyperharmonicEulerSum k r (p + 1) a =
      (-1) ^ p * iteratedDeriv p (fun b => iteratedDeriv k
        (fun s => Complex.Gamma b * Complex.Gamma (1 - s) * (Complex.Gamma (b + 1 - s))⁻¹) r) a

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Theorem 1.

Uses 1

Statement usesGeneralized hyperharmonic Euler sum

TheoremPolygamma structure of hyperharmonic Euler sums#●ready to prove

Let \(k \ge 0\), \(r\) and \(p\) be integers with \(1 \le r \le p\), and let \(a \in \mathbb{C} \setminus \mathbb{Z}\). Write \(H_n^{[r]}(k)\) for the generalized hyperharmonic number. Then the series \(S(k, r; p + 1; a) = \sum_{n \ge k} H_n^{[r]}(k) / (n + a)^{p+1}\) converges absolutely, and its sum lies in the algebra generated over the field \(\mathbb{Q}(a)\) by \(\psi(a) + \gamma\), the derivatives \(\psi^{(j)}(a)\) for \(j \ge 1\) and the zeta values \(\zeta(j)\) for \(j \ge 2\), where \(\psi\) is the digamma function and \(\gamma\) is the Euler–Mascheroni constant.

The source also extends the result by continuity to integers \(a > -k\); that case is not included here.

public theorem_wanted hyperharmonicEulerSum_mem_adjoin_digamma (k p r : ℕ) (hr : 1 ≤ r)
    (hrp : r ≤ p) (a : ℂ) (ha : ∀ m : ℤ, a ≠ m) :
    Summable (fun n : ℕ =>
        generalizedHyperharmonic (n + k) k r / (((n + k : ℕ) : ℂ) + a) ^ ((p : ℤ) + 1)) ∧
      hyperharmonicEulerSum k r (p + 1) a ∈
        Algebra.adjoin (IntermediateField.adjoin ℚ {a})
          ({Complex.digamma a + Real.eulerMascheroniConstant} ∪
            Set.range (fun j : ℕ => iteratedDeriv (j + 1) Complex.digamma a) ∪
            Set.range fun j : ℕ => riemannZeta (j + 2))

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Theorem 2.

Uses 2

Statement usesGeneralized hyperharmonic numbers · Generalized hyperharmonic Euler sum

TheoremHyperharmonic Euler sums with rational parameters#●ready to prove

Let \(a\) and \(r\) be rational numbers that are not integers, with \(a - r \in \mathbb{Z}\), let \(N \ge 1\) be an integer with \(N a \in \mathbb{Z}\), and let \(k \ge 0\) and \(p > r\) be integers. Put \(M = \mathrm{lcm}(2N, 4)\) and \(\omega = e^{2 \pi i / M}\), a primitive \(M\)-th root of unity. Write \(H_n^{[r]}(k)\) for the generalized hyperharmonic numbers. Then the generalized hyperharmonic Euler sum \(S(k, r; p; a) = \sum_{n \ge k} H_n^{[r]}(k) / (n + a)^p\) converges, and \(S(k, r; p; a) = \pi x\) for some \(x\) in the algebra generated over the field \(\mathbb{Q}(\omega + \omega^{-1})\) by \(\psi(a) + \gamma\), the derivatives \(\psi^{(j)}(a)\) for \(j \ge 1\), \(\pi^2\) and the odd zeta values \(\zeta(3), \zeta(5), \dots\), where \(\psi\) is the digamma function and \(\gamma\) is the Euler–Mascheroni constant.

The field \(\mathbb{Q}(\omega + \omega^{-1})\) does not depend on which primitive \(M\)-th root of unity \(\omega\) is chosen.

public theorem_wanted hyperharmonicEulerSum_rat_mem_pi_mul_adjoin (a r : ℚ)
    (ha : ∀ m : ℤ, a ≠ m) (hr : ∀ m : ℤ, r ≠ m) (N : ℕ) (hN : 0 < N)
    (hNa : ∃ m : ℤ, N * a = m) (har : ∃ m : ℤ, a - r = m) (k : ℕ) (p : ℤ) (hp : r < p) :
    let ζ : ℂ := Complex.exp (2 * π * Complex.I / (Nat.lcm (2 * N) 4 : ℂ))
    Summable (fun n : ℕ =>
        generalizedHyperharmonic (n + k) k r / (((n + k : ℕ) : ℂ) + a) ^ p) ∧
      ∃ x ∈ Algebra.adjoin (IntermediateField.adjoin ℚ {ζ + ζ⁻¹})
          ({Complex.digamma a + Real.eulerMascheroniConstant} ∪
            Set.range (fun j : ℕ => iteratedDeriv (j + 1) Complex.digamma a) ∪
            {(π : ℂ) ^ 2} ∪ Set.range fun j : ℕ => riemannZeta (2 * j + 3)),
        hyperharmonicEulerSum k r p a = π * x

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Theorem 3.

Uses 2

Statement usesGeneralized hyperharmonic numbers · Generalized hyperharmonic Euler sum

TheoremDuality for hyperharmonic Euler sums#●ready to prove

Let \(k, p \ge 0\) be integers and let \(r, t\) be complex numbers with \(\operatorname{Re}(r) < p + 1\) and \(\operatorname{Re}(t) < k + 1\). Write \(H_n^{[r]}(k)\) for the generalized hyperharmonic number. Then the series \(S(k, r; p + 1; 1 - t) = \sum_{n \ge k} H_n^{[r]}(k) / (n + 1 - t)^{p+1}\) and \(S(p, t; k + 1; 1 - r) = \sum_{n \ge p} H_n^{[t]}(p) / (n + 1 - r)^{k+1}\) both converge absolutely, and \(S(k, r; p + 1; 1 - t) = S(p, t; k + 1; 1 - r)\).

public theorem_wanted hyperharmonicEulerSum_duality (k p : ℕ) (r t : ℂ) (hr : r.re < p + 1)
    (ht : t.re < k + 1) :
    Summable (fun n : ℕ =>
        generalizedHyperharmonic (n + k) k r / (((n + k : ℕ) : ℂ) + (1 - t)) ^ ((p : ℤ) + 1)) ∧
      Summable (fun n : ℕ =>
        generalizedHyperharmonic (n + p) p t / (((n + p : ℕ) : ℂ) + (1 - r)) ^ ((k : ℤ) + 1)) ∧
      hyperharmonicEulerSum k r (p + 1) (1 - t) = hyperharmonicEulerSum p t (k + 1) (1 - r)

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 1.

Uses 2

Statement usesGeneralized hyperharmonic numbers · Generalized hyperharmonic Euler sum

TheoremHyperharmonic Euler sums with integer parameter and half-integer shift#●ready to prove

Let \(k \ge 0\), \(r\) and \(p\) be integers with \(1 \le r < p\), and let \(a \in \frac{1}{2} + \mathbb{Z}\). Write \(H_n^{[r]}(k)\) for the generalized hyperharmonic number. Then the series \(S(k, r; p; a) = \sum_{n \ge k} H_n^{[r]}(k) / (n + a)^p\) converges absolutely, and its sum lies in the \(\mathbb{Q}\)-subalgebra \(\mathbb{Q}[\log 2, \pi^2, \zeta(2j+1)_{j \ge 1}]\) of \(\mathbb{C}\) generated by \(\log 2\), \(\pi^2\) and the odd zeta values.

The source prints \(1 \le r \le p\), but at \(r = p\) the series can diverge (for \(k = 0\), \(r = p = 1\) and \(a = 1/2\) it is \(\sum_{n \ge 0} 1/(n + 1/2)\)), and the source's Theorem 2 has exponent \(p + 1\) for \(r \le p\); so \(r < p\) is assumed.

public theorem_wanted hyperharmonicEulerSum_half_mem_eulerSumPeriodAlgebra (k p r : ℕ)
    (hr : 1 ≤ r) (hrp : r < p) (c : ℤ) :
    Summable (fun n : ℕ =>
        generalizedHyperharmonic (n + k) k r / (((n + k : ℕ) : ℂ) + (c + 1 / 2)) ^ (p : ℤ)) ∧
      hyperharmonicEulerSum k r p (c + 1 / 2) ∈ eulerSumPeriodAlgebra

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 2, first part.

Uses 3

Statement usesThe algebra generated by \(\log 2\), \(\pi^2\) and odd zeta values · Generalized hyperharmonic numbers · Generalized hyperharmonic Euler sum

TheoremOdd harmonic central binomial sums at integer shifts#●ready to prove

Let \(k \ge 0\) and \(p > 0\) be integers, let \(a\) be an integer with \(a > -k\), and let \(O_n(k)\) denote the depth-\(k\) odd harmonic number. Then the series of real numbers

\[\sum_{n \ge 0} \frac{\binom{2n}{n} O_n(k)}{4^n (n + a)^p}\]

converges, and its sum lies in the algebra \(\mathbb{Q}[\log 2, \pi^2, \zeta(3), \zeta(5), \dots]\) viewed inside \(\mathbb{C}\). The terms with \(n < k\) are \(0\), since \(O_n(k) = 0\) for \(n < k\), so this is also the sum over \(n \ge k\).

public theorem_wanted centralBinom_oddMultipleHarmonic_sum_mem_eulerSumPeriodAlgebra
    (k p : ℕ) (hp : 0 < p) (a : ℤ) (ha : -(k : ℤ) < a) :
    Summable (fun n : ℕ =>
        ((2 * n).choose n : ℝ) * oddMultipleHarmonic k n / (4 ^ n * ((n : ℝ) + a) ^ p)) ∧
      ((∑' n : ℕ, ((2 * n).choose n : ℝ) * oddMultipleHarmonic k n /
          (4 ^ n * ((n : ℝ) + a) ^ p) : ℝ) : ℂ) ∈ eulerSumPeriodAlgebra

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 2, second part.

Uses 2

Statement usesThe algebra generated by \(\log 2\), \(\pi^2\) and odd zeta values · Odd multiple harmonic numbers

TheoremCentral binomial odd harmonic sum with denominator \(4^n (n + 1)^2\)#●ready to prove

For every integer \(k \ge 0\), with \(O_n(k)\) the depth-\(k\) odd harmonic number,

\[\sum_{n \ge 0} \frac{\binom{2n}{n} O_n(k)}{4^n (n + 1)^2} = 2^{k+2} (k + 1 - \log 2) - \sum_{j=1}^{k} (2^{k+2} - 2^j) \zeta(k + 2 - j).\]

The terms with \(n < k\) vanish, since \(O_n(k) = 0\) for \(n < k\).

public theorem_wanted centralBinom_oddMultipleHarmonic_div_succ_sq_tsum (k : ℕ) :
    ((∑' n : ℕ, ((2 * n).choose n : ℝ) * oddMultipleHarmonic k n /
        (4 ^ n * ((n : ℝ) + 1) ^ 2) : ℝ) : ℂ) =
      2 ^ (k + 2) * (k + 1 - Real.log 2) -
        ∑ j ∈ Finset.Icc 1 k, (2 ^ (k + 2) - 2 ^ j) * riemannZeta ((k + 2 - j : ℕ) : ℂ)

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 3.

Uses 1

Statement usesOdd multiple harmonic numbers

TheoremOdd harmonic central binomial sums with weight \(\binom{n}{m}\) at integer shifts#●ready to prove

Let \(k, m \ge 0\) and \(p > m\) be integers, let \(a\) be an integer with \(a > -k\), and let \(O_n(k)\) denote the depth-\(k\) odd harmonic number. Then the series of real numbers

\[\sum_{n \ge 0} \binom{n}{m} \frac{\binom{2n}{n} O_n(k)}{4^n (n + a)^p}\]

converges, and its sum lies in the algebra \(\mathbb{Q}[\log 2, \pi^2, \zeta(3), \zeta(5), \dots]\) viewed inside \(\mathbb{C}\). The terms with \(n < k\) are \(0\), since \(O_n(k) = 0\) for \(n < k\), so this is also the sum over \(n \ge k\).

public theorem_wanted choose_centralBinom_oddMultipleHarmonic_sum_mem_eulerSumPeriodAlgebra
    (k p m : ℕ) (hmp : m < p) (a : ℤ) (ha : -(k : ℤ) < a) :
    Summable (fun n : ℕ => (n.choose m : ℝ) * ((2 * n).choose n : ℝ) *
        oddMultipleHarmonic k n / (4 ^ n * ((n : ℝ) + a) ^ p)) ∧
      ((∑' n : ℕ, (n.choose m : ℝ) * ((2 * n).choose n : ℝ) * oddMultipleHarmonic k n /
          (4 ^ n * ((n : ℝ) + a) ^ p) : ℝ) : ℂ) ∈ eulerSumPeriodAlgebra

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 4, first part.

Uses 2

Statement usesThe algebra generated by \(\log 2\), \(\pi^2\) and odd zeta values · Odd multiple harmonic numbers

TheoremCentral binomial odd harmonic sums over \(\binom{n+m}{m}\) lie in the period algebra#●ready to prove

Let \(O_n(k)\) denote the depth-\(k\) odd harmonic number and let \(A = \mathbb{Q}[\log 2, \pi^2, \zeta(3), \zeta(5), \ldots] \subseteq \mathbb{C}\) be the Euler sum period algebra. For integers \(k \ge 0\), \(p \ge 1\), \(m \ge 1\) and \(a > -k\), the series

\[\sum_{n \ge 0} \frac{\binom{2n}{n} O_n(k)}{4^n \binom{n+m}{m} (n + a)^p}\]

converges and its sum lies in \(A\).

The terms with \(n < k\) are \(0\) because \(O_n(k) = 0\) there; this includes the term with \(n + a = 0\) when \(a \le 0\).

public theorem_wanted centralBinom_oddMultipleHarmonic_div_choose_sum_mem_eulerSumPeriodAlgebra
    (k p m : ℕ) (hp : 0 < p) (hm : 0 < m) (a : ℤ) (ha : -(k : ℤ) < a) :
    Summable (fun n : ℕ => ((2 * n).choose n : ℝ) * oddMultipleHarmonic k n /
        (4 ^ n * ((n + m).choose m : ℝ) * ((n : ℝ) + a) ^ p)) ∧
      ((∑' n : ℕ, ((2 * n).choose n : ℝ) * oddMultipleHarmonic k n /
          (4 ^ n * ((n + m).choose m : ℝ) * ((n : ℝ) + a) ^ p) : ℝ) : ℂ) ∈
        eulerSumPeriodAlgebra

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 4, second part.

Uses 2

Statement usesThe algebra generated by \(\log 2\), \(\pi^2\) and odd zeta values · Odd multiple harmonic numbers

TheoremCatalan–odd harmonic sum with denominator \(4^n n\)#●ready to prove

Let \(k \ge 1\) be an integer, let \(C_n\) denote the \(n\)-th Catalan number and \(O_n(k)\) the depth-\(k\) odd harmonic number. Then

\[\sum_{n \ge 1} \frac{C_n O_n(k)}{4^n n} = (2^{k+1} - 1) \zeta(k + 1) - 2^{k+1}.\]

Since \(O_n(k) = 0\) for \(n < k\), the sum can equally be taken over \(n \ge k\).

public theorem_wanted catalan_oddMultipleHarmonic_div_tsum (k : ℕ) (hk : 0 < k) :
    ((∑' n : ℕ, (catalan n : ℝ) * oddMultipleHarmonic k n / (4 ^ n * n) : ℝ) : ℂ) =
      (2 ^ (k + 1) - 1) * riemannZeta (k + 1) - 2 ^ (k + 1)

Sources

• the denominator \(4^n n\) is read from the record's two-row layout (\(C_n O_n(k)\) over \(4^n n\)).
• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 5.

Uses 1

Statement usesOdd multiple harmonic numbers

TheoremQuarter-shift hyperharmonic Euler sums#●ready to prove

Let \(k \ge 0\) and \(0 \le r < p\) be integers, and let \(a\) be a complex number with \(a \in \tfrac14 + \mathbb{Z}\) or \(a \in \tfrac34 + \mathbb{Z}\). Write \(H_n^{[r]}(k)\) for the generalized hyperharmonic numbers. Then the generalized hyperharmonic Euler sum \(S(k, r; p; a) = \sum_{n \ge k} H_n^{[r]}(k) / (n + a)^p\) converges, and its value lies in the \(\mathbb{Q}\)-algebra \(\mathbb{Q}[\log 2, \pi, \zeta(3), \zeta(5), \dots, \beta(2), \beta(4), \dots]\) of complex numbers, where \(\beta(s) = \sum_{n \ge 0} (-1)^n / (2n+1)^s\) is the Dirichlet beta function.

The source prints \(0 \le r \le p\); at \(r = p \ge 1\) the series diverges (for \(k = 0\) and \(r = p = 1\) it is \(\sum_{n \ge 0} 1 / (n + a)\)), so \(r < p\) is used.

public theorem_wanted hyperharmonicEulerSum_quarter_mem_eulerSumBetaPeriodAlgebra
    (k p r : ℕ) (hrp : r < p) (a : ℂ) (ha : ∃ c : ℤ, a = c + 1 / 4 ∨ a = c + 3 / 4) :
    Summable (fun n : ℕ =>
        generalizedHyperharmonic (n + k) k r / (((n + k : ℕ) : ℂ) + a) ^ (p : ℤ)) ∧
      hyperharmonicEulerSum k r p a ∈ eulerSumBetaPeriodAlgebra

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 6.

Uses 3

Statement usesThe algebra generated by \(\log 2\), \(\pi\), odd zeta values and even beta values · Generalized hyperharmonic numbers · Generalized hyperharmonic Euler sum

TheoremHyperharmonic Euler sums with half-integer parameter and shift#●ready to prove

Let \(k \ge 0\) be an integer, let \(r, a \in \frac{1}{2} + \mathbb{Z}\), and let \(p\) be an integer with \(p > r\). Write \(H_n^{[r]}(k)\) for the generalized hyperharmonic number. Then the series \(S(k, r; p; a) = \sum_{n \ge k} H_n^{[r]}(k) / (n + a)^p\) converges absolutely, and its sum equals \(\pi x\) for some \(x\) in the \(\mathbb{Q}\)-subalgebra \(\mathbb{Q}[\log 2, \pi^2, \zeta(2j+1)_{j \ge 1}]\) of \(\mathbb{C}\) generated by \(\log 2\), \(\pi^2\) and the odd zeta values.

public theorem_wanted hyperharmonicEulerSum_half_half_mem_pi_mul (k : ℕ) (c d p : ℤ)
    (hp : c < p) :
    Summable (fun n : ℕ => generalizedHyperharmonic (n + k) k (c + 1 / 2) /
        (((n + k : ℕ) : ℂ) + (d + 1 / 2)) ^ p) ∧
      ∃ x ∈ eulerSumPeriodAlgebra,
        hyperharmonicEulerSum k (c + 1 / 2) p (d + 1 / 2) = π * x

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 7, first part.

Uses 3

Statement usesThe algebra generated by \(\log 2\), \(\pi^2\) and odd zeta values · Generalized hyperharmonic numbers · Generalized hyperharmonic Euler sum

TheoremOdd harmonic central binomial sums with weight \(\binom{n}{m}\) at half-integer shifts#●ready to prove

Let \(k, m \ge 0\) and \(p > m\) be integers, let \(a = d + 1/2\) with \(d\) an integer, and let \(O_n(k)\) denote the depth-\(k\) odd harmonic number. Then the series of real numbers

\[\sum_{n \ge 0} \binom{n}{m} \frac{\binom{2n}{n} O_n(k)}{4^n (n + a)^p}\]

converges, and its sum equals \(\pi x\) for some \(x\) in the algebra \(\mathbb{Q}[\log 2, \pi^2, \zeta(3), \zeta(5), \dots] \subseteq \mathbb{C}\). The terms with \(n < k\) are \(0\), since \(O_n(k) = 0\) for \(n < k\), so this is also the sum over \(n \ge k\).

public theorem_wanted choose_centralBinom_oddMultipleHarmonic_half_sum_mem_pi_mul
    (k p m : ℕ) (hmp : m < p) (d : ℤ) :
    Summable (fun n : ℕ => (n.choose m : ℝ) * ((2 * n).choose n : ℝ) *
        oddMultipleHarmonic k n / (4 ^ n * ((n : ℝ) + (d + 1 / 2)) ^ p)) ∧
      ∃ x ∈ eulerSumPeriodAlgebra,
        ((∑' n : ℕ, (n.choose m : ℝ) * ((2 * n).choose n : ℝ) * oddMultipleHarmonic k n /
          (4 ^ n * ((n : ℝ) + (d + 1 / 2)) ^ p) : ℝ) : ℂ) = π * x

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 7, second part.

Uses 2

Statement usesThe algebra generated by \(\log 2\), \(\pi^2\) and odd zeta values · Odd multiple harmonic numbers

TheoremCentral binomial odd harmonic sums over \(\binom{n+m}{m}\) at half-integer shifts#●ready to prove

Let \(O_n(k)\) denote the depth-\(k\) odd harmonic number and let \(A = \mathbb{Q}[\log 2, \pi^2, \zeta(3), \zeta(5), \ldots] \subseteq \mathbb{C}\) be the Euler sum period algebra. For integers \(k \ge 0\), \(p \ge 1\), \(m \ge 1\) and \(d\), put \(a = d + 1/2\). Then the series

\[\sum_{n \ge 0} \frac{\binom{2n}{n} O_n(k)}{4^n \binom{n+m}{m} (n + a)^p}\]

converges, and its sum equals \(q + \pi x\) for some rational number \(q\) and some \(x \in A\).

The terms with \(n < k\) vanish, since \(O_n(k) = 0\) for \(n < k\).

public theorem_wanted centralBinom_oddMultipleHarmonic_div_choose_half_sum_mem
    (k p m : ℕ) (hp : 0 < p) (hm : 0 < m) (d : ℤ) :
    Summable (fun n : ℕ => ((2 * n).choose n : ℝ) * oddMultipleHarmonic k n /
        (4 ^ n * ((n + m).choose m : ℝ) * ((n : ℝ) + (d + 1 / 2)) ^ p)) ∧
      ∃ q : ℚ, ∃ x ∈ eulerSumPeriodAlgebra,
        ((∑' n : ℕ, ((2 * n).choose n : ℝ) * oddMultipleHarmonic k n /
          (4 ^ n * ((n + m).choose m : ℝ) * ((n : ℝ) + (d + 1 / 2)) ^ p) : ℝ) : ℂ) =
          q + π * x

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 7, third part.

Uses 2

Statement usesThe algebra generated by \(\log 2\), \(\pi^2\) and odd zeta values · Odd multiple harmonic numbers

TheoremCatalan–odd harmonic sums at the shift \(1/2\)#●ready to prove

Let \(k \ge 0\) and \(p \ge 1\) be integers, let \(C_n\) denote the \(n\)-th Catalan number and \(O_n(k)\) the depth-\(k\) odd harmonic number, and let \(A = \mathbb{Q}[\log 2, \pi^2, \zeta(3), \zeta(5), \ldots] \subseteq \mathbb{C}\) be the Euler sum period algebra. Then the series

\[S = \sum_{n \ge 0} \frac{C_n O_n(k)}{4^n (n + 1/2)^p}\]

converges, and \(S - (-2)^p 2^{k+1} = \pi x\) for some \(x \in A\).

The terms with \(n < k\) vanish, since \(O_n(k) = 0\) for \(n < k\).

public theorem_wanted catalan_oddMultipleHarmonic_half_sum_sub_mem_pi_mul (k p : ℕ)
    (hp : 0 < p) :
    Summable (fun n : ℕ => (catalan n : ℝ) * oddMultipleHarmonic k n /
        (4 ^ n * ((n : ℝ) + 1 / 2) ^ p)) ∧
      ∃ x ∈ eulerSumPeriodAlgebra,
        ((∑' n : ℕ, (catalan n : ℝ) * oddMultipleHarmonic k n /
          (4 ^ n * ((n : ℝ) + 1 / 2) ^ p) : ℝ) : ℂ) - (-2) ^ p * 2 ^ (k + 1) = π * x

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 8.

Uses 2

Statement usesThe algebra generated by \(\log 2\), \(\pi^2\) and odd zeta values · Odd multiple harmonic numbers

TheoremCentral binomial sum over \(\binom{n+m}{m} (n + 1/2)^2\)#●ready to prove

Let \(m \ge 1\) be an integer and write \(O_m = O_m(1) = \sum_{i=1}^{m} 1/(i - 1/2)\) for the depth-one odd harmonic number. Then

\[\sum_{n \ge 0} \frac{\binom{2n}{n}}{4^n \binom{n+m}{m} (n + 1/2)^2} = \frac{4^m \pi}{\binom{2m}{m}} \bigl(2 \log 2 - O_m\bigr) - \sum_{q=1}^{m} \frac{(-4)^q \binom{m}{q}}{\binom{2q}{q} (q - 1/2)^2}.\]

public theorem_wanted centralBinom_div_choose_half_sq_tsum (m : ℕ) (hm : 0 < m) :
    ∑' n : ℕ, ((2 * n).choose n : ℝ) / (4 ^ n * ((n + m).choose m : ℝ) * ((n : ℝ) + 1 / 2) ^ 2) =
      4 ^ m * π / ((2 * m).choose m : ℝ) * (2 * Real.log 2 - oddMultipleHarmonic 1 m) -
        ∑ q ∈ Finset.Icc 1 m,
          (-4 : ℝ) ^ q * (m.choose q : ℝ) / (((2 * q).choose q : ℝ) * ((q : ℝ) - 1 / 2) ^ 2)

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 9, first identity.

Uses 1

Statement usesOdd multiple harmonic numbers

TheoremOdd harmonic central binomial sum over \(\binom{n+m}{m} (n + 1/2)^2\)#●ready to prove

Let \(m \ge 1\) be an integer and write \(O_j = O_j(1) = \sum_{i=1}^{j} 1/(i - 1/2)\) for the depth-one odd harmonic number. Then

\[\sum_{n \ge 0} \frac{\binom{2n}{n} (O_n - O_m)}{4^n \binom{n+m}{m} (n + 1/2)^2} = \frac{4^m \pi}{\binom{2m}{m}} \Bigl(\bigl(O_m - 2 \log 2\bigr)^2 - \frac{\pi^2}{6}\Bigr) + \sum_{q=1}^{m} \frac{(-4)^q \binom{m}{q} (O_m - O_q)}{\binom{2q}{q} (q - 1/2)^2}.\]

Here \(\pi^2 / 6 = \zeta(2)\).

public theorem_wanted centralBinom_oddHarmonic_sub_div_choose_half_sq_tsum (m : ℕ)
    (hm : 0 < m) :
    ∑' n : ℕ, ((2 * n).choose n : ℝ) * (oddMultipleHarmonic 1 n - oddMultipleHarmonic 1 m) /
        (4 ^ n * ((n + m).choose m : ℝ) * ((n : ℝ) + 1 / 2) ^ 2) =
      4 ^ m * π / ((2 * m).choose m : ℝ) *
          ((oddMultipleHarmonic 1 m - 2 * Real.log 2) ^ 2 - π ^ 2 / 6) +
        ∑ q ∈ Finset.Icc 1 m,
          (-4 : ℝ) ^ q * (m.choose q : ℝ) * (oddMultipleHarmonic 1 m - oddMultipleHarmonic 1 q) /
            (((2 * q).choose q : ℝ) * ((q : ℝ) - 1 / 2) ^ 2)

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 9, second identity.

Uses 1

Statement usesOdd multiple harmonic numbers

TheoremProducts of quarter-shift hyperharmonic Euler sums#●ready to prove

Let \(k_1, k_2 \ge 0\) be integers, let \(r_1, r_2 \in \tfrac12 + \mathbb{Z}\), \(a \in \tfrac14 + \mathbb{Z}\) and \(b \in \tfrac34 + \mathbb{Z}\), and let \(p_1, p_2\) be integers with \(p_1 > r_1\) and \(p_2 > r_2\). Write \(H_n^{[r]}(k)\) for the generalized hyperharmonic numbers and \(S(k, r; p; a) = \sum_{n \ge k} H_n^{[r]}(k) / (n + a)^p\) for the generalized hyperharmonic Euler sum. Then the series \(S(k_1, r_1; p_1; a)\) and \(S(k_2, r_2; p_2; b)\) converge, and \(S(k_1, r_1; p_1; a) \, S(k_2, r_2; p_2; b) = \pi x\) for some \(x\) in the \(\mathbb{Q}\)-algebra \(\mathbb{Q}[\log 2, \pi, \zeta(3), \zeta(5), \dots, \beta(2), \beta(4), \dots]\) of complex numbers, where \(\beta(s) = \sum_{n \ge 0} (-1)^n / (2n+1)^s\) is the Dirichlet beta function.

public theorem_wanted hyperharmonicEulerSum_quarter_mul_mem_pi_mul (k₁ k₂ : ℕ)
    (c₁ c₂ d e p₁ p₂ : ℤ) (h₁ : c₁ < p₁) (h₂ : c₂ < p₂) :
    Summable (fun n : ℕ => generalizedHyperharmonic (n + k₁) k₁ (c₁ + 1 / 2) /
        (((n + k₁ : ℕ) : ℂ) + (d + 1 / 4)) ^ p₁) ∧
      Summable (fun n : ℕ => generalizedHyperharmonic (n + k₂) k₂ (c₂ + 1 / 2) /
        (((n + k₂ : ℕ) : ℂ) + (e + 3 / 4)) ^ p₂) ∧
      ∃ x ∈ eulerSumBetaPeriodAlgebra,
        hyperharmonicEulerSum k₁ (c₁ + 1 / 2) p₁ (d + 1 / 4) *
          hyperharmonicEulerSum k₂ (c₂ + 1 / 2) p₂ (e + 3 / 4) = π * x

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 11, first part.

Uses 3

Statement usesThe algebra generated by \(\log 2\), \(\pi\), odd zeta values and even beta values · Generalized hyperharmonic numbers · Generalized hyperharmonic Euler sum

TheoremProducts of quarter-shift central binomial odd harmonic sums#●ready to prove

Let \(O_n(k)\) denote the depth-\(k\) odd harmonic number, let \(\beta(s) = \sum_{n \ge 0} (-1)^n / (2n+1)^s\) be the Dirichlet beta function, and let \(B = \mathbb{Q}[\log 2, \pi, \zeta(3), \zeta(5), \ldots, \beta(2), \beta(4), \ldots] \subseteq \mathbb{C}\). For integers \(k_1, k_2 \ge 0\) and \(p_1, p_2 \ge 1\), the series

\[S_1 = \sum_{n \ge 0} \frac{\binom{2n}{n} O_n(k_1)}{4^n (n + 1/4)^{p_1}}, \qquad S_2 = \sum_{n \ge 0} \frac{\binom{2n}{n} O_n(k_2)}{4^n (n + 3/4)^{p_2}}\]

both converge, and \(S_1 S_2 = \pi x\) for some \(x \in B\).

The terms with \(n < k_i\) vanish, since \(O_n(k_i) = 0\) for \(n < k_i\).

public theorem_wanted centralBinom_oddMultipleHarmonic_quarter_mul_mem_pi_mul
    (k₁ k₂ p₁ p₂ : ℕ) (hp₁ : 0 < p₁) (hp₂ : 0 < p₂) :
    Summable (fun n : ℕ => ((2 * n).choose n : ℝ) * oddMultipleHarmonic k₁ n /
        (4 ^ n * ((n : ℝ) + 1 / 4) ^ p₁)) ∧
      Summable (fun n : ℕ => ((2 * n).choose n : ℝ) * oddMultipleHarmonic k₂ n /
        (4 ^ n * ((n : ℝ) + 3 / 4) ^ p₂)) ∧
      ∃ x ∈ eulerSumBetaPeriodAlgebra,
        ((∑' n : ℕ, ((2 * n).choose n : ℝ) * oddMultipleHarmonic k₁ n /
            (4 ^ n * ((n : ℝ) + 1 / 4) ^ p₁) : ℝ) : ℂ) *
          ((∑' n : ℕ, ((2 * n).choose n : ℝ) * oddMultipleHarmonic k₂ n /
            (4 ^ n * ((n : ℝ) + 3 / 4) ^ p₂) : ℝ) : ℂ) = π * x

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 11, second part.

Uses 2

Statement usesThe algebra generated by \(\log 2\), \(\pi\), odd zeta values and even beta values · Odd multiple harmonic numbers

TheoremOdd harmonic central binomial sums with weight \(\binom{n}{m}\) at shift \(1/4\)#●ready to prove

Let \(k, m \ge 0\) and \(p > m\) be integers, let \(O_n(k)\) denote the depth-\(k\) odd harmonic number, let \(\beta\) be the Dirichlet beta function, and let

\[\varpi = 2 \int_0^1 \frac{dt}{\sqrt{1 - t^4}}\]

be the lemniscate constant. Then the series of real numbers

\[\sum_{n \ge 0} \binom{n}{m} \frac{\binom{2n}{n} O_n(k)}{4^n (n + 1/4)^p}\]

converges, and its sum equals \(\varpi x\) for some \(x\) in the algebra \(\mathbb{Q}[\log 2, \pi, \zeta(2j+1)_{j \ge 1}, \beta(2j)_{j \ge 1}] \subseteq \mathbb{C}\). The terms with \(n < k\) are \(0\), since \(O_n(k) = 0\) for \(n < k\), so this is also the sum over \(n \ge k\).

public theorem_wanted choose_centralBinom_oddMultipleHarmonic_quarter_sum_mem
    (k p m : ℕ) (hmp : m < p) :
    Summable (fun n : ℕ => (n.choose m : ℝ) * ((2 * n).choose n : ℝ) *
        oddMultipleHarmonic k n / (4 ^ n * ((n : ℝ) + 1 / 4) ^ p)) ∧
      ∃ x ∈ eulerSumBetaPeriodAlgebra,
        ((∑' n : ℕ, (n.choose m : ℝ) * ((2 * n).choose n : ℝ) * oddMultipleHarmonic k n /
          (4 ^ n * ((n : ℝ) + 1 / 4) ^ p) : ℝ) : ℂ) =
          (((2 * ∫ t in (0 : ℝ)..1, (Real.sqrt (1 - t ^ 4))⁻¹) : ℝ) : ℂ) * x

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 10, first statement.

Uses 2

Statement usesThe algebra generated by \(\log 2\), \(\pi\), odd zeta values and even beta values · Odd multiple harmonic numbers

TheoremOdd harmonic central binomial sums with weight \(\binom{n}{m}\) at shift \(3/4\)#●ready to prove

Let \(k, m \ge 0\) and \(p > m\) be integers, let \(O_n(k)\) denote the depth-\(k\) odd harmonic number, let \(\beta\) be the Dirichlet beta function, and let

\[\varpi = 2 \int_0^1 \frac{dt}{\sqrt{1 - t^4}}\]

be the lemniscate constant. Then the series of real numbers

\[\sum_{n \ge 0} \binom{n}{m} \frac{\binom{2n}{n} O_n(k)}{4^n (n + 3/4)^p}\]

converges, and its sum equals \((\pi / \varpi) x\) for some \(x\) in the algebra \(\mathbb{Q}[\log 2, \pi, \zeta(2j+1)_{j \ge 1}, \beta(2j)_{j \ge 1}] \subseteq \mathbb{C}\). The terms with \(n < k\) are \(0\), since \(O_n(k) = 0\) for \(n < k\), so this is also the sum over \(n \ge k\).

public theorem_wanted choose_centralBinom_oddMultipleHarmonic_three_quarter_sum_mem
    (k p m : ℕ) (hmp : m < p) :
    Summable (fun n : ℕ => (n.choose m : ℝ) * ((2 * n).choose n : ℝ) *
        oddMultipleHarmonic k n / (4 ^ n * ((n : ℝ) + 3 / 4) ^ p)) ∧
      ∃ x ∈ eulerSumBetaPeriodAlgebra,
        ((∑' n : ℕ, (n.choose m : ℝ) * ((2 * n).choose n : ℝ) * oddMultipleHarmonic k n /
          (4 ^ n * ((n : ℝ) + 3 / 4) ^ p) : ℝ) : ℂ) =
          ((π / (2 * ∫ t in (0 : ℝ)..1, (Real.sqrt (1 - t ^ 4))⁻¹) : ℝ) : ℂ) * x

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 10, second statement.

Uses 2

Statement usesThe algebra generated by \(\log 2\), \(\pi\), odd zeta values and even beta values · Odd multiple harmonic numbers

TheoremCentral binomial odd harmonic sums over \(\binom{n+m}{m}\) at the shift \(1/4\)#●ready to prove

Let \(O_n(k)\) denote the depth-\(k\) odd harmonic number, let \(\beta(s) = \sum_{n \ge 0} (-1)^n / (2n+1)^s\) be the Dirichlet beta function, let \(B = \mathbb{Q}[\log 2, \pi, \zeta(3), \zeta(5), \ldots, \beta(2), \beta(4), \ldots] \subseteq \mathbb{C}\), and let \(\varpi = 2 \int_0^1 dt / \sqrt{1 - t^4}\) be the lemniscate constant. For integers \(k \ge 0\), \(p \ge 1\) and \(m \ge 1\), the series

\[\sum_{n \ge 0} \frac{\binom{2n}{n} O_n(k)}{4^n \binom{n+m}{m} (n + 1/4)^p}\]

converges, and its sum equals \(q + \varpi x\) for some rational number \(q\) and some \(x \in B\).

The terms with \(n < k\) vanish, since \(O_n(k) = 0\) for \(n < k\).

public theorem_wanted centralBinom_oddMultipleHarmonic_div_choose_quarter_sum_mem
    (k p m : ℕ) (hp : 0 < p) (hm : 0 < m) :
    Summable (fun n : ℕ => ((2 * n).choose n : ℝ) * oddMultipleHarmonic k n /
        (4 ^ n * ((n + m).choose m : ℝ) * ((n : ℝ) + 1 / 4) ^ p)) ∧
      ∃ q : ℚ, ∃ x ∈ eulerSumBetaPeriodAlgebra,
        ((∑' n : ℕ, ((2 * n).choose n : ℝ) * oddMultipleHarmonic k n /
          (4 ^ n * ((n + m).choose m : ℝ) * ((n : ℝ) + 1 / 4) ^ p) : ℝ) : ℂ) =
          q + (((2 * ∫ t in (0 : ℝ)..1, (Real.sqrt (1 - t ^ 4))⁻¹) : ℝ) : ℂ) * x

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 10, third statement.

Uses 2

Statement usesThe algebra generated by \(\log 2\), \(\pi\), odd zeta values and even beta values · Odd multiple harmonic numbers

TheoremCentral binomial odd harmonic sums over \(\binom{n+m}{m}\) at the shift \(3/4\)#●ready to prove

Let \(O_n(k)\) denote the depth-\(k\) odd harmonic number, let \(\beta(s) = \sum_{n \ge 0} (-1)^n / (2n+1)^s\) be the Dirichlet beta function, let \(B = \mathbb{Q}[\log 2, \pi, \zeta(3), \zeta(5), \ldots, \beta(2), \beta(4), \ldots] \subseteq \mathbb{C}\), and let \(\varpi = 2 \int_0^1 dt / \sqrt{1 - t^4}\) be the lemniscate constant. For integers \(k \ge 0\), \(p \ge 1\) and \(m \ge 1\), the series

\[\sum_{n \ge 0} \frac{\binom{2n}{n} O_n(k)}{4^n \binom{n+m}{m} (n + 3/4)^p}\]

converges, and its sum equals \(q + (\pi / \varpi) x\) for some rational number \(q\) and some \(x \in B\).

The terms with \(n < k\) vanish, since \(O_n(k) = 0\) for \(n < k\).

public theorem_wanted centralBinom_oddMultipleHarmonic_div_choose_three_quarter_sum_mem
    (k p m : ℕ) (hp : 0 < p) (hm : 0 < m) :
    Summable (fun n : ℕ => ((2 * n).choose n : ℝ) * oddMultipleHarmonic k n /
        (4 ^ n * ((n + m).choose m : ℝ) * ((n : ℝ) + 3 / 4) ^ p)) ∧
      ∃ q : ℚ, ∃ x ∈ eulerSumBetaPeriodAlgebra,
        ((∑' n : ℕ, ((2 * n).choose n : ℝ) * oddMultipleHarmonic k n /
          (4 ^ n * ((n + m).choose m : ℝ) * ((n : ℝ) + 3 / 4) ^ p) : ℝ) : ℂ) =
          q + ((π / (2 * ∫ t in (0 : ℝ)..1, (Real.sqrt (1 - t ^ 4))⁻¹) : ℝ) : ℂ) * x

Sources

• Paul Thomas Young, Parametric Euler Sums of Generalized Hyperharmonic Numbers, INTEGERS 26 (2026), Article A81, Corollary 10, fourth statement.

Uses 2

Statement usesThe algebra generated by \(\log 2\), \(\pi\), odd zeta values and even beta values · Odd multiple harmonic numbers

15 fully proved23 ready to prove

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