Generalized Stirling numbers

Deformations and generalizations of Stirling numbers: r-Stirling, associated Stirling, Hsu–Shiue, Jacobi–Stirling, Legendre–Stirling, meromorphic and degenerate Stirling numbers, α-Stirling and generalized Stirling matrices, and (q,r)-Whitney and Dowling numbers.

DefinitionGeneralized Stirling numbers of the first kind#✓fully proved

For natural numbers \(n\), \(m\) and a real number \(x\), the generalized Stirling number of the first kind is

\[{n \brack m}_x = \sum_{i=0}^{n-m} \binom{i+m}{m} {n \brack i+m} x^i,\]

where \({n \brack k}\) is the unsigned Stirling number of the first kind, the number of permutations of \(n\) elements with exactly \(k\) cycles. For \(m > n\) the sum consists of the single term \(i = 0\), which is \({n \brack m} = 0\), so \({n \brack m}_x = 0\).

noncomputable def generalizedStirlingFirst (n m : ℕ) (x : ℝ) : ℝ :=
  Finset.sum (Finset.range (n - m + 1)) (fun i =>
    (Nat.choose (i + m) m : ℝ) * (Nat.stirlingFirst n (i + m) : ℝ) * x ^ i)
Sources
  • Carlitz and Broder, as recorded in Cereceda, arXiv:2401.13696v4, lines 132-142; Axioms 14 (2025), 746.
Used by 1
Used bySum formula for generalized Stirling numbers of the first kind

DefinitionGeneralized Stirling numbers of the second kind#✓fully proved

For natural numbers \(n\), \(m\) and a real number \(x\), the generalized Stirling number of the second kind is

\[{n \brace m}_x = \sum_{i=0}^{n-m} \binom{n}{i} {n-i \brace m} x^i,\]

where \({n \brace k}\) is the Stirling number of the second kind, the number of partitions of an \(n\)-element set into \(k\) nonempty blocks. For \(m > n\) the sum consists of the single term \(i = 0\), which is \(\binom{n}{0} {n \brace m} = 0\), so \({n \brace m}_x = 0\).

noncomputable def generalizedStirlingSecond (n m : ℕ) (x : ℝ) : ℝ :=
  Finset.sum (Finset.range (n - m + 1)) (fun i =>
    (Nat.choose n i : ℝ) * (Nat.stirlingSecond (n - i) m : ℝ) * x ^ i)
Sources
  • Carlitz and Broder, as recorded in Cereceda, arXiv:2401.13696v4, lines 132-142; Axioms 14 (2025), 746.
Used by 3
Used bySum formula for generalized Stirling numbers of the second kind · r-Stirling numbers as generalized Stirling numbers · Expansion by generalized Stirling numbers of the second kind

TheoremSum formula for generalized Stirling numbers of the first kind#✓fully proved

For natural numbers \(m \le n\) and every real number \(x\), the generalized Stirling number of the first kind satisfies

\[{n \brack m}_x = \sum_{i=0}^{n-m} \binom{i+m}{m} {n \brack i+m} x^i,\]

where \({n \brack k}\) is the unsigned Stirling number of the first kind; the sum includes both endpoints \(i = 0\) and \(i = n - m\).

theorem generalizedStirlingFirst_eq_sum (n m : ℕ) (x : ℝ) (_h : m ≤ n) :
    generalizedStirlingFirst n m x =
      Finset.sum (Finset.range (n - m + 1)) (fun i =>
        (Nat.choose (i + m) m : ℝ) * (Nat.stirlingFirst n (i + m) : ℝ) * x ^ i)
Sources
  • Cereceda, arXiv:2401.13696v4, lines 132-142.
Uses 1
Statement usesGeneralized Stirling numbers of the first kind

TheoremSum formula for generalized Stirling numbers of the second kind#✓fully proved

For natural numbers \(m \le n\) and every real number \(x\), the generalized Stirling number of the second kind satisfies

\[{n \brace m}_x = \sum_{i=0}^{n-m} \binom{n}{i} {n-i \brace m} x^i,\]

where \({n \brace k}\) is the Stirling number of the second kind; the sum includes both endpoints \(i = 0\) and \(i = n - m\).

theorem generalizedStirlingSecond_eq_sum (n m : ℕ) (x : ℝ) (_h : m ≤ n) :
    generalizedStirlingSecond n m x =
      Finset.sum (Finset.range (n - m + 1)) (fun i =>
        (Nat.choose n i : ℝ) * (Nat.stirlingSecond (n - i) m : ℝ) * x ^ i)
Sources
  • Cereceda, arXiv:2401.13696v4, lines 132-142.
Uses 1
Statement usesGeneralized Stirling numbers of the second kind
4 fully proved

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