Generalized Bell polynomials

DefinitionGeneralized Bell numbers#✓fully proved

For a commutative ring \(R\), elements \(s, h \in R\) and a natural number \(n\), the generalized Bell number is

\[\mathfrak{B}_{s;h}(n) = \sum_{k=0}^{n} \mathfrak{S}_{s;h}(n, k),\]

where the generalized Stirling numbers \(\mathfrak{S}_{s;h}(a, j) \in R\) are defined for all natural numbers \(a\) and \(j\) by \(\mathfrak{S}_{s;h}(0, 0) = 1\), \(\mathfrak{S}_{s;h}(0, j + 1) = \mathfrak{S}_{s;h}(a + 1, 0) = 0\) and \(\mathfrak{S}_{s;h}(a + 1, j + 1) = \mathfrak{S}_{s;h}(a, j) + h\,(j + 1 + s \max(a - j - 1, 0))\,\mathfrak{S}_{s;h}(a, j + 1)\); here the natural number \(\max(a - j - 1, 0)\) is viewed as an element of \(R\).

def genBellNumber {R : Type*} [CommRing R] (s h : R) (n : ℕ) : R :=
  ∑ k ∈ Finset.range (n + 1), genStirling s h n k

Used by 1

Used byGeneralized Bell polynomials at one

DefinitionGeneralized Bell polynomials#✓fully proved

For a commutative ring \(R\), elements \(s, h \in R\), a natural number \(n\) and \(x \in R\), the generalized Bell polynomial evaluated at \(x\) is

\[\mathfrak{B}_{s;h|n}(x) = \sum_{k=0}^{n} \mathfrak{S}_{s;h}(n, k)\, x^k,\]

where the generalized Stirling numbers \(\mathfrak{S}_{s;h}(a, j) \in R\) are defined for all natural numbers \(a\) and \(j\) by \(\mathfrak{S}_{s;h}(0, 0) = 1\), \(\mathfrak{S}_{s;h}(0, j + 1) = \mathfrak{S}_{s;h}(a + 1, 0) = 0\) and \(\mathfrak{S}_{s;h}(a + 1, j + 1) = \mathfrak{S}_{s;h}(a, j) + h\,(j + 1 + s \max(a - j - 1, 0))\,\mathfrak{S}_{s;h}(a, j + 1)\); here the natural number \(\max(a - j - 1, 0)\) is viewed as an element of \(R\).

def genBellPolynomial {R : Type*} [CommRing R] (s h : R) (n : ℕ) (x : R) : R :=
  ∑ k ∈ Finset.range (n + 1), genStirling s h n k * x ^ k

Used by 1

Used byGeneralized Bell polynomials at one

TheoremRecurrence for generalized Stirling numbers#✓fully proved

Let \(R\) be a commutative ring and \(s, h \in R\). The generalized Stirling numbers \(\mathfrak{S}_{s;h}(n, k) \in R\) are defined by \(\mathfrak{S}_{s;h}(0, 0) = 1\), \(\mathfrak{S}_{s;h}(0, k + 1) = \mathfrak{S}_{s;h}(n + 1, 0) = 0\) and the recurrence below. For all natural numbers \(n\) and \(k\),

\[\mathfrak{S}_{s;h}(n + 1, k + 1) = \mathfrak{S}_{s;h}(n, k) + h\,(k + 1 + s \max(n - k - 1, 0))\,\mathfrak{S}_{s;h}(n, k + 1).\]

Here \(\max(n - k - 1, 0)\) is a natural number viewed as an element of \(R\).

theorem genStirling_succ_succ {R : Type*} [CommRing R] (s h : R) (n k : ℕ) :
    genStirling s h (n + 1) (k + 1) = genStirling s h n k +
      h * ((↑(k + 1) : R) + s * (↑(n - (k + 1)) : R)) *
        genStirling s h n (k + 1)

TheoremGeneralized Bell polynomials at one#✓fully proved

For a commutative ring \(R\), elements \(s, h \in R\) and a natural number \(n\), the generalized Bell polynomial \(\mathfrak{B}_{s;h|n}\) evaluated at \(x = 1\) equals the generalized Bell number \(\mathfrak{B}_{s;h}(n)\).

theorem genBellPolynomial_eval_one {R : Type*} [CommRing R] (s h : R) (n : ℕ) :
    genBellPolynomial s h n 1 = genBellNumber s h n

Uses 2

Statement usesGeneralized Bell numbers · Generalized Bell polynomials

4 fully proved

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