Sums of reciprocals of binomial coefficients

TheoremFrisch's alternating reciprocal-binomial identity#✓fully proved

For natural numbers \(n\), \(b\) and \(c\) with \(0 < c \le b\), the rational numbers \(\sum_{k=0}^{n} (-1)^k \binom{n}{k} \binom{b+k}{c}^{-1}\) and \(\frac{c}{n+c} \binom{n+b}{b-c}^{-1}\) are equal.

theorem frisch_alternating_reciprocal_binomial
    (n b c : ℕ) (hc : 0 < c) (hcb : c ≤ b) :
    ∑ k ∈ Finset.range (n + 1),
      (-1 : ℚ) ^ k * (Nat.choose n k : ℚ) * (Nat.choose (b + k) c : ℚ)⁻¹ =
      (c : ℚ) / (n + c) * (Nat.choose (n + b) (b - c) : ℚ)⁻¹
Sources
  • H. W. Gould and Jocelyn Quaintance, "On the Binomial Identities of Frisch and Klamkin," Journal of Integer Sequences 19 (2016), who attribute the identity to Ragnar Frisch, "Sur les semi-invariants et moments employés dans l'étude des distributions statistiques" (1926); lines 92–95 of https://cs.uwaterloo.ca/journals/JIS/VOL19/Gould/gould8.tex. The inverse Frisch identity, Abel's infinite variant and the Klamkin identities are not included.

TheoremKlamkin's binomial reciprocal identity#✓fully proved

For natural numbers \(n\), \(b\) and \(x\) with \(n + b \le x\), the following equality of rational numbers holds:

\[\sum_{k=0}^{n} \frac{\binom{n}{k}}{\binom{x}{k+b}} = \frac{x+1}{(x-n+1)\binom{x-n}{b}}.\]
theorem klamkin_binomial_reciprocal_identity
    (n b x : ℕ) (h : n + b ≤ x) :
    ∑ k ∈ Finset.range (n + 1),
        (Nat.choose n k : ℚ) / (Nat.choose x (k + b) : ℚ) =
      ((x + 1 : ℕ) : ℚ) /
        (((x - n + 1 : ℕ) : ℚ) * (Nat.choose (x - n) b : ℚ))
Sources
2 fully proved

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