Frisch and Klamkin binomial identities

TheoremGauss summation at Frisch's specialization#✓fully proved

Let \(n \ge 1\) be an integer and let \(b\) and \(c\) be real numbers with \(c > n\) and \(b > -1\). Then

\[ {}_2F_1(1 + b - c, n; b + 1; 1) = \frac{\Gamma(c - n)\,\Gamma(b + 1)}{\Gamma(c)\,\Gamma(b - n + 1)}, \]

where \({}_2F_1(\alpha, \beta; \gamma; 1) = \sum_{k \ge 0} \frac{(\alpha)_k (\beta)_k}{(\gamma)_k\, k!}\) and \((x)_k\) is the rising factorial.

The Gamma function is taken to be \(0\) at its poles and division by \(0\) gives \(0\), so the right side is \(0\) when \(b - n + 1\) is a nonpositive integer.

theorem gauss_hypergeometric_summation_frisch (n : ℕ) (hn : 1 ≤ n) (b c : ℝ)
    (hcn : (n : ℝ) < c) (hb : -1 < b) :
    ordinaryHypergeometric (1 + b - c) (n : ℝ) (b + 1) (1 : ℝ) =
      Real.Gamma (c - n) * Real.Gamma (b + 1) /
        (Real.Gamma c * Real.Gamma (b - n + 1))
Sources

TheoremKlamkin's Gauss summation#✓fully proved

Let \(n \ge 1\) be an integer and let \(a\) and \(b\) be real numbers such that \(-a\) and \(-b\) are not nonnegative integers, \(a + 1 + n < 0\), and \(a - b + 1\) is not a nonnegative integer. Then

\[ {}_2F_1(n, b + 1; b - a; 1) = \frac{\Gamma(-a - n - 1)\,\Gamma(b - a)}{\Gamma(b - a - n)\,\Gamma(-a - 1)}, \]

where \({}_2F_1(\alpha, \beta; \gamma; 1) = \sum_{k \ge 0} \frac{(\alpha)_k (\beta)_k}{(\gamma)_k\, k!}\) and \((x)_k\) is the rising factorial.

The condition \(a + 1 + n < 0\) makes the series converge. The Gamma function is taken to be \(0\) at its poles and division by \(0\) gives \(0\), so the right side is \(0\) when \(b - a - n\) is a nonpositive integer.

theorem gauss_hypergeometric_summation_klamkin (n : ℕ) (a b : ℝ)
    (ha : ∀ k : ℕ, -a ≠ (k : ℝ))
    (hb : ∀ k : ℕ, -b ≠ (k : ℝ))
    (hn : 1 ≤ n)
    (hconv : a + 1 + n < 0)
    (hadmissible : ∀ k : ℕ, a - b + 1 ≠ (k : ℝ)) :
    ordinaryHypergeometric (n : ℝ) (b + 1) (b - a) (1 : ℝ) =
      Real.Gamma (-a - n - 1) * Real.Gamma (b - a) /
        (Real.Gamma (b - a - n) * Real.Gamma (-a - 1))
Sources
  • U. Abel, "A Short Proof of the Binomial Identities of Frisch and Klamkin," Journal of Integer Sequences 23 (2020), Article 20.7.1.
Used by 1
Used byKlamkin's Gauss summation as a convergent series

TheoremKlamkin's Gauss summation as a convergent series#✓fully proved

Let \(n \ge 1\) be an integer and let \(a\) and \(b\) be real numbers such that \(-a\) and \(-b\) are not nonnegative integers, \(a + 1 + n < 0\), and \(a - b + 1\) is not a nonnegative integer. Then the series

\[ \sum_{k \ge 0} \frac{(n)_k\,(b + 1)_k}{(b - a)_k\, k!} \]

converges absolutely, with sum

\[ \frac{\Gamma(-a - n - 1)\,\Gamma(b - a)}{\Gamma(b - a - n)\,\Gamma(-a - 1)}, \]

where \((x)_k = x(x + 1) \cdots (x + k - 1)\) is the rising factorial.

The Gamma function is taken to be \(0\) at its poles and division by \(0\) gives \(0\), so this sum is \(0\) when \(b - a - n\) is a nonpositive integer.

theorem hasSum_ordinaryHypergeometricCoefficient_klamkin (n : ℕ) (a b : ℝ)
    (ha : ∀ k : ℕ, -a ≠ (k : ℝ))
    (hb : ∀ k : ℕ, -b ≠ (k : ℝ))
    (hn : 1 ≤ n)
    (hconv : a + 1 + n < 0)
    (hadmissible : ∀ k : ℕ, a - b + 1 ≠ (k : ℝ)) :
    HasSum (fun k => ordinaryHypergeometricCoefficient (n : ℝ) (b + 1) (b - a) k)
      (Real.Gamma (-a - n - 1) * Real.Gamma (b - a) /
        (Real.Gamma (b - a - n) * Real.Gamma (-a - 1)))
Uses 1 · Used by 1
Proof usesKlamkin's Gauss summation
Used byAbel's variant of Klamkin's identity

TheoremAbel's variant of Klamkin's identity#✓fully proved

For real numbers \(c\) and \(d\) write \(\binom{c}{d} = \Gamma(c+1) / (\Gamma(d+1) \, \Gamma(c-d+1))\); for natural numbers this is the usual binomial coefficient. Let \(a\) and \(b\) be real numbers such that \(-a\) and \(-b\) are not in \(\{0, 1, 2, \dots\}\) and \(a - b\) is not an integer, and let \(n \ge 1\) be a natural number with \(n < -a - 1\). Then the series

\[\sum_{k \ge 0} (-1)^k \binom{n+k-1}{k} \binom{a}{b+k}^{-1}\]

converges unconditionally to \(\frac{a+1}{a+n+1} \binom{a+n}{b}^{-1}\).

Abel assumes only \(a - b + 1 \notin \mathbb{N}\); here \(a - b\) must avoid every integer, since otherwise some \(\Gamma(a - b - k + 1)\) is evaluated at a pole of \(\Gamma\).

Proof: Rewrite the summand as an ordinary hypergeometric coefficient and apply Abel's specialization of Gauss's formula (Abramowitz-Stegun 15.1.20), then simplify by Gamma recurrence.

public theorem variant_Klamkin (a b : ℝ) (n : ℕ)
    (h_neg_a : ∀ m : ℕ, -a ≠ (m : ℝ))
    (h_neg_b : ∀ m : ℕ, -b ≠ (m : ℝ))
    (h_n_pos : 1 ≤ n)
    (h_n_bound : (n : ℝ) < -a - 1)
    (h_ab : ∀ m : ℤ, a - b ≠ (m : ℝ)) :
    HasSum (fun k : ℕ => (-1 : ℝ) ^ k * (Nat.choose (n + k - 1) k : ℝ) *
      (Real.Gamma (a + 1) /
        (Real.Gamma (b + (k : ℝ) + 1) *
          Real.Gamma (a - (b + (k : ℝ)) + 1)))⁻¹)
      ((a + 1) / (a + (n : ℝ) + 1) *
        ((Real.Gamma (a + (n : ℝ) + 1) /
          (Real.Gamma (b + 1) * Real.Gamma (a + (n : ℝ) - b + 1)))⁻¹))
Sources
  • Ulrich Abel, "A Short Proof of the Binomial Identities of Frisch and Klamkin," Journal of Integer Sequences 23 (2020), Article 20.7.1, Theorem, equation (Identity-variant-Klamkin), lines 210–218, https://cs.uwaterloo.ca/journals/JIS/VOL23/Abel/abel12.tex
Uses 1
Proof usesKlamkin's Gauss summation as a convergent series
4 fully proved

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