Odd harmonic numbers

DefinitionOdd harmonic numbers#✓fully proved

For a natural number \(n\), the \(n\)-th odd harmonic number is the rational number \(O_n = \sum_{m=1}^{n} \frac{1}{2m-1}\), the sum of the reciprocals of the first \(n\) odd positive integers; in particular \(O_0 = 0\).

def oddHarmonic (n : ℕ) : ℚ :=
  ∑ m ∈ range n, 1 / (2 * (m : ℚ) + 1)
Sources
  • Kunle Adegoke and Robert Frontczak, Three Classes of Combinatorial Sums Involving Central Binomial Coefficients and Harmonic Numbers, INTEGERS 26 (2026), Article A109, Section 1 (definition of O_n).
Used by 11
Used byOdd harmonic central binomial sums via r-Stirling numbers · Odd harmonic central binomial sums for powers one to three · Recurrence for central binomial sums with odd harmonic numbers · Alternating reciprocal central binomial sum with \(O_{n+1} - O_{k+1}\) · Alternating reciprocal central binomial sum with \(O_n - O_k\) · Central binomial sums weighted by \(1 - O_{k-1}\), in r-Stirling form · Power-weighted convolution of \(4^k\) with \(\binom{2m}{m} O_m\) · Power-weighted convolution of \(4^k\) with \(\binom{2m}{m} (1 - O_{m-1})/(2m-1)\) · Odd harmonic numbers in terms of harmonic numbers · Recurrence for the odd harmonic numbers · The zeroth odd harmonic number

TheoremThe zeroth odd harmonic number#✓fully proved

The odd harmonic number \(O_0\) equals \(0\).

@[simp]
theorem oddHarmonic_zero : oddHarmonic 0 = 0
Sources
  • Kunle Adegoke and Robert Frontczak, Three Classes of Combinatorial Sums Involving Central Binomial Coefficients and Harmonic Numbers, INTEGERS 26 (2026), Article A109, Section 1.
Uses 1 · Used by 1
Statement usesOdd harmonic numbers
Used byOdd harmonic numbers in terms of harmonic numbers

TheoremRecurrence for the odd harmonic numbers#✓fully proved

For every natural number \(n\), the odd harmonic numbers satisfy \(O_{n+1} = O_n + \frac{1}{2n+1}\).

theorem oddHarmonic_succ (n : ℕ) :
    oddHarmonic (n + 1) = oddHarmonic n + 1 / (2 * (n : ℚ) + 1)
Sources
  • Kunle Adegoke and Robert Frontczak, Three Classes of Combinatorial Sums Involving Central Binomial Coefficients and Harmonic Numbers, INTEGERS 26 (2026), Article A109, Section 1 (used in the Remark after Corollary 1).
Uses 1 · Used by 1
Statement usesOdd harmonic numbers
Used byOdd harmonic numbers in terms of harmonic numbers

TheoremOdd harmonic numbers in terms of harmonic numbers#✓fully proved

For every natural number \(n\), the odd harmonic number \(O_n\) and the harmonic numbers \(H_k = \sum_{j=1}^{k} \frac{1}{j}\) satisfy \(O_n = H_{2n} - \frac{H_n}{2}\).

theorem oddHarmonic_eq_harmonic (n : ℕ) : oddHarmonic n = harmonic (2 * n) - harmonic n / 2
Sources
  • Kunle Adegoke and Robert Frontczak, Three Classes of Combinatorial Sums Involving Central Binomial Coefficients and Harmonic Numbers, INTEGERS 26 (2026), Article A109, Section 1 (definition of O_n).
Uses 3
Statement usesOdd harmonic numbers
Proof usesRecurrence for the odd harmonic numbers · The zeroth odd harmonic number
4 fully proved

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