Hsu–Shiue generalized Stirling numbers

TheoremFactorization of the generalized Stirling matrix#✓fully proved

Let \(S\) assign to real numbers \(a, b, c\) and natural numbers \(n, k\) a real number \(S(a, b, c; n, k)\), such that for all real \(a, b, c, x\) and every natural number \(n\)

\[\prod_{i=0}^{n-1} (x - i a) = \sum_{k=0}^{n} S(a, b, c; n, k) \prod_{i=0}^{k-1} (x - c - i b),\]

and \(S(a, b, c; n, k) = 0\) whenever \(n < k\). Then for all real \(\alpha, \beta, \gamma\) and all natural numbers \(n, k\)

\[S(\alpha, \beta, \gamma; n, k) = \sum_{j=0}^{n} \sum_{l=0}^{j} S(\alpha, 0, 0; n, j)\, S(0, 0, \gamma; j, l)\, S(0, \beta, 0; l, k).\]

In matrix form, the generalized Stirling matrix with parameters \((\alpha, \beta, \gamma)\) is the product of the matrices for \((\alpha, 0, 0)\) (first kind), \((0, 0, \gamma)\) (translation) and \((0, \beta, 0)\) (second kind).

theorem generalized_stirling_matrix_factorization
    (S : ℝ → ℝ → ℝ → ℕ → ℕ → ℝ)
    (hS : ∀ (a b c : ℝ) (n : ℕ) (x : ℝ),
      (∏ i ∈ Finset.range n, (x - (i : ℝ) * a)) =
        ∑ k ∈ Finset.range (n + 1), S a b c n k *
          ∏ i ∈ Finset.range k, ((x - c) - (i : ℝ) * b))
    (hzero : ∀ (a b c : ℝ) (n k : ℕ), n < k → S a b c n k = 0)
    (α β γ : ℝ) (n k : ℕ) :
    S α β γ n k =
      ∑ j ∈ Finset.range (n + 1), ∑ l ∈ Finset.range (j + 1),
        S α 0 0 n j * S 0 0 γ j l * S 0 β 0 l k
Sources
  • Jiaqiang Pan, “Matrix Decomposition of the Unified Generalized Stirling Numbers and Inversion of the Generalized Factorial Matrices,” Journal of Integer Sequences 15 (2012), Article 12.6.6, Theorem t:Decomposition, lines 250-256, https://cs.uwaterloo.ca/journals/JIS/VOL15/Pan/pan19.tex.

DefinitionGeneralized falling factorial#✓fully proved

Let \(K\) be \(\mathbb{R}\) or \(\mathbb{C}\). For \(z, \alpha \in K\) the generalized falling factorial with increment \(\alpha\) is the sequence defined by \((z \mid \alpha)_0 = 1\) and \((z \mid \alpha)_{n+1} = (z \mid \alpha)_n \, (z - n\alpha)\) for \(n \ge 0\), so that \((z \mid \alpha)_n = z (z - \alpha) \cdots (z - (n-1)\alpha)\) for \(n \ge 1\).

public noncomputable def generalizedFallingFactorial (z alpha : K) : Nat → K
  | 0 => 1
  | n + 1 => generalizedFallingFactorial z alpha n * (z - (n : K) * alpha)
Used by 7
Used byDegenerate Bernoulli numbers · The zeroth degenerate Bernoulli number · Degenerate exponential series · Degenerate harmonic numbers of rank and degenerate Stirling numbers of the first kind · Alternating degenerate exponential sum of degenerate harmonic numbers of rank · Shifted alternating degenerate exponential sum of degenerate harmonic numbers of rank · Characterization of the Hsu–Shiue generalized Stirling numbers

DefinitionCharacterization of the Hsu–Shiue generalized Stirling numbers#✓fully proved

Let \(K\) be \(\mathbb{R}\) or \(\mathbb{C}\) and let \(\alpha, \beta, r \in K\). For \(z \in K\) write \((z \mid \alpha)_0 = 1\) and \((z \mid \alpha)_n = z (z - \alpha) \cdots (z - (n-1)\alpha)\) for \(n \ge 1\) (the generalized falling factorial). An array \((S(n,k))_{n, k \ge 0}\) in \(K\) is a Hsu–Shiue family with parameters \((\alpha, \beta, r)\) when \(\alpha\), \(\beta\) and \(r\) are not all zero, \(S(n, 0) = (r \mid \alpha)_n\) for every \(n \ge 0\), \(S(n, k) = 0\) whenever \(n < k\), and

\[S(n+1, k+1) = S(n, k) + \bigl((k+1)\beta - n\alpha + r\bigr) S(n, k+1)\]

for all \(n, k \ge 0\).

The last condition is the source recurrence \(S(n+1, k) = S(n, k-1) + (k\beta - n\alpha + r) S(n, k)\) at the indices \(k \ge 1\); the initial column and the triangular support fix the remaining values.

public def IsHsuShiueFamily (S : Nat → Nat → K) (alpha beta r : K) : Prop :=
  ¬ (alpha = 0 ∧ beta = 0 ∧ r = 0) ∧
    (∀ n, S n 0 = generalizedFallingFactorial r alpha n) ∧
      (∀ n k, n < k → S n k = 0) ∧
        ∀ n k, S (n + 1) (k + 1) =
          S n k + (((k : K) + 1) * beta - (n : K) * alpha + r) * S n (k + 1)
Uses 1 · Used by 1
Statement usesGeneralized falling factorial
Used byScherk's identity with \(s\)-rising factorials
3 fully proved

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