Central binomial and Catalan–harmonic generating functions

TheoremGenerating function of Catalan–harmonic products#✓fully proved

Let \(C_n\) be the \(n\)th Catalan number and \(H_n = \sum_{k=1}^{n} 1/k\) the \(n\)th harmonic number, so that \(H_0 = 0\). For every real number \(x\) with \(|x| < 1/4\),

\[ \sum_{n=0}^{\infty} C_n H_n x^{n+1} = \log 2 + \sqrt{1-4x}\,\log\bigl(2\sqrt{1-4x}\bigr) - \bigl(1 + \sqrt{1-4x}\bigr)\log\bigl(1 + \sqrt{1-4x}\bigr). \]

theorem catalan_harmonic_generating_function_canonical
    (x : ℝ) (hx : |x| < 1 / 4) :
    (∑' n : ℕ, (catalan n : ℝ) * ((harmonic n : ℚ) : ℝ) * x ^ (n + 1)) =
      Real.log 2 +
        Real.sqrt (1 - 4 * x) * Real.log (2 * Real.sqrt (1 - 4 * x)) -
        (1 + Real.sqrt (1 - 4 * x)) * Real.log (1 + Real.sqrt (1 - 4 * x))

Used by 1

Used byCatalan–harmonic generating function with explicit coefficients

TheoremCatalan–harmonic generating function with explicit coefficients#✓fully proved

For every real number \(x\) with \(|x| < 1/4\),

\[ \sum_{n=1}^{\infty} \frac{\binom{2n}{n}}{n+1} \Bigl(\sum_{k=1}^{n} \frac{1}{k}\Bigr) x^{n+1} = \log 2 + \sqrt{1-4x}\,\log\bigl(2\sqrt{1-4x}\bigr) - \bigl(1 + \sqrt{1-4x}\bigr)\log\bigl(1 + \sqrt{1-4x}\bigr). \]

Here \(\binom{2n}{n}/(n+1)\) is the \(n\)th Catalan number (the division is exact) and \(\sum_{k=1}^{n} 1/k\) is the \(n\)th harmonic number.

theorem catalan_harmonic_generating_function
    (x : ℝ) (hx : |x| < 1 / 4) :
    (∑' n : ℕ,
      if n = 0 then 0
      else
        (((Nat.choose (2 * n) n / (n + 1) : ℕ) : ℝ) *
          (∑ k ∈ Finset.range n, (((k + 1 : ℕ) : ℝ)⁻¹)) * x ^ (n + 1))) =
      Real.log 2 +
        Real.sqrt (1 - 4 * x) * Real.log (2 * Real.sqrt (1 - 4 * x)) -
        (1 + Real.sqrt (1 - 4 * x)) * Real.log (1 + Real.sqrt (1 - 4 * x))

Sources

• Hongwei Chen, "Interesting Series Associated with Central Binomial Coefficients, Catalan Numbers and Harmonic Numbers," Journal of Integer Sequences 19 (2016), Article 16.1.5, Corollary, lines 156–160, https://cs.uwaterloo.ca/journals/JIS/VOL19/Chen/chen21.tex; obtained there by integrating the central-binomial harmonic generating function.

Uses 1

Proof usesGenerating function of Catalan–harmonic products

TheoremCentral binomial–harmonic generating function#✓fully proved

Let \(H_n = \sum_{k=1}^{n} 1/k\) be the \(n\)th harmonic number. For every real number \(x\) with \(|x| < 1/4\), the series \(\sum_{n=1}^{\infty} \binom{2n}{n} H_n x^n\) converges unconditionally, with sum

\[ \sum_{n=1}^{\infty} \binom{2n}{n} H_n x^n = \frac{2}{\sqrt{1-4x}} \log \frac{1 + \sqrt{1-4x}}{2\sqrt{1-4x}}. \]

theorem hasSum_centralBinomial_harmonic (x : ℝ) (hx : |x| < 1 / 4) :
  HasSum
    (fun n : ℕ => (Nat.choose (2 * (n + 1)) (n + 1) : ℝ) * (harmonic (n + 1) : ℝ) *
      x ^ (n + 1))
    (2 / Real.sqrt (1 - 4 * x) *
      Real.log ((1 + Real.sqrt (1 - 4 * x)) / (2 * Real.sqrt (1 - 4 * x))))

Sources

• Hongwei Chen, "Interesting Series Associated with Central Binomial Coefficients, Catalan Numbers and Harmonic Numbers," Journal of Integer Sequences 19 (2016), Article 16.1.5, Theorem, lines 147–151, https://cs.uwaterloo.ca/journals/JIS/VOL19/Chen/chen21.tex.

3 fully proved

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