Laguerre–Stirling–Bell connection pair

TheoremLaguerre–Stirling–Bell connection pair#✓fully proved

Let \(\alpha\), \(\beta\) and \(x\) be real numbers, and let \(n\) and \(r\) be natural numbers. Write \(\binom{k}{j}\) for the binomial coefficients, \(c(n,k)\) for the unsigned Stirling numbers of the first kind, \(S(n,k)\) for the Stirling numbers of the second kind, and \(y^{\overline{m}} = y(y+1)\cdots(y+m-1)\) for the rising factorial. For natural numbers \(m\), \(s\) and real \(y\), let \(B_m(y) = \sum_{l=0}^{m} S(m,l)\, y^l\) be the Bell polynomial and

\[L_{m,s}(y) = \sum_{k=0}^{m} \frac{1}{k!} \sum_{j=0}^{k} (-1)^{k-j} \binom{k}{j} (\alpha s + \beta j)^{\overline{m}}\, y^k\]

the generalized Laguerre polynomial. Then

\[L_{n,r}(x) = \sum_{k=0}^{n} \sum_{j=0}^{k} c(n,k) \binom{k}{j} (\alpha r)^{k-j} \beta^j B_j(x)\]

and

\[\sum_{k=0}^{n} (-1)^{n-k} S(n,k)\, L_{k,r}(x) = \sum_{k=0}^{n} \binom{n}{k} (\alpha r)^{n-k} \beta^k B_k(x).\]

MetaMathlibExt.stirling_bell_polynomial_connection_pair_general · 21 lines

theorem stirling_bell_polynomial_connection_pair_general
    (α β x : ℝ) (n r : ℕ) :
    let bellPolynomial : ℕ → ℝ → ℝ := fun m y ↦
      ∑ l ∈ Finset.range (m + 1), (Nat.stirlingSecond m l : ℝ) * y ^ l
    let generalizedLaguerrePolynomial : ℕ → ℕ → ℝ → ℝ := fun m s y ↦
      ∑ k ∈ Finset.range (m + 1),
        ((∑ j ∈ Finset.range (k + 1),
            (-1 : ℝ) ^ (k - j) * (Nat.choose k j : ℝ) *
              ∏ i ∈ Finset.range m,
                (α * (s : ℝ) + β * (j : ℝ) + (i : ℝ))) /
          (Nat.factorial k : ℝ)) * y ^ k
    generalizedLaguerrePolynomial n r x =
        ∑ k ∈ Finset.range (n + 1), ∑ j ∈ Finset.range (k + 1),
          (Nat.stirlingFirst n k : ℝ) * (Nat.choose k j : ℝ) *
            (α * (r : ℝ)) ^ (k - j) * β ^ j * bellPolynomial j x ∧
      ∑ k ∈ Finset.range (n + 1),
          (-1 : ℝ) ^ (n - k) * (Nat.stirlingSecond n k : ℝ) *
            generalizedLaguerrePolynomial k r x =
        ∑ k ∈ Finset.range (n + 1),
          (Nat.choose n k : ℝ) * (α * (r : ℝ)) ^ (n - k) * β ^ k *
            bellPolynomial k x

Sources

• Mark Shattuck, "Combinatorial Properties of a Generalized Class of Laguerre Polynomials," Journal of Integer Sequences 24 (2021), Article 21.2.1, Theorem, equations, lines 330–339, https://cs.uwaterloo.ca/journals/JIS/VOL24/Shattuck/shattuck24.tex

Uses 1 · Used by 1

Proof usesExplicit formula for Stirling numbers of the second kind

Used byLaguerre–Stirling–Bell connection pair, source form

TheoremLaguerre–Stirling–Bell connection pair, source form#✓fully proved

Let \(\alpha\), \(\beta\) and \(x\) be real numbers with \(\beta \neq 0\), and let \(n\) and \(r\) be natural numbers. Write \(\binom{k}{j}\) for the binomial coefficients, \(c(n,k)\) for the unsigned Stirling numbers of the first kind, \(S(n,k)\) for the Stirling numbers of the second kind, and \(y^{\overline{m}} = y(y+1)\cdots(y+m-1)\) for the rising factorial. For natural numbers \(m\), \(s\) and real \(y\), let \(B_m(y) = \sum_{l=0}^{m} S(m,l)\, y^l\) be the Bell polynomial and

\[L_{m,s}(y) = \sum_{k=0}^{m} \frac{1}{k!} \sum_{j=0}^{k} (-1)^{k-j} \binom{k}{j} (\alpha s + \beta j)^{\overline{m}}\, y^k\]

the generalized Laguerre polynomial. Then

\[L_{n,r}(x) = \sum_{k=0}^{n} \sum_{j=0}^{k} c(n,k) \binom{k}{j} (\alpha r)^{k-j} \beta^j B_j(x)\]

and

\[\sum_{k=0}^{n} (-1)^{n-k} S(n,k)\, L_{k,r}(x) = \sum_{k=0}^{n} \binom{n}{k} (\alpha r)^{n-k} \beta^k B_k(x).\]

The hypothesis \(\beta \neq 0\) is not used: the same two identities hold for every real \(\beta\).

MetaMathlibExt.stirling_bell_polynomial_connection_pair · 22 lines

@[nolint unusedArguments]
theorem stirling_bell_polynomial_connection_pair
    (α β x : ℝ) (n r : ℕ) (hβ : β ≠ 0) :
    let bellPolynomial : ℕ → ℝ → ℝ := fun m y ↦
      ∑ l ∈ Finset.range (m + 1), (Nat.stirlingSecond m l : ℝ) * y ^ l
    let generalizedLaguerrePolynomial : ℕ → ℕ → ℝ → ℝ := fun m s y ↦
      ∑ k ∈ Finset.range (m + 1),
        ((∑ j ∈ Finset.range (k + 1),
            (-1 : ℝ) ^ (k - j) * (Nat.choose k j : ℝ) *
              ∏ i ∈ Finset.range m,
                (α * (s : ℝ) + β * (j : ℝ) + (i : ℝ))) /
          (Nat.factorial k : ℝ)) * y ^ k
    generalizedLaguerrePolynomial n r x =
        ∑ k ∈ Finset.range (n + 1), ∑ j ∈ Finset.range (k + 1),
          (Nat.stirlingFirst n k : ℝ) * (Nat.choose k j : ℝ) *
            (α * (r : ℝ)) ^ (k - j) * β ^ j * bellPolynomial j x ∧
      ∑ k ∈ Finset.range (n + 1),
          (-1 : ℝ) ^ (n - k) * (Nat.stirlingSecond n k : ℝ) *
            generalizedLaguerrePolynomial k r x =
        ∑ k ∈ Finset.range (n + 1),
          (Nat.choose n k : ℝ) * (α * (r : ℝ)) ^ (n - k) * β ^ k *
            bellPolynomial k x

Sources

• Mark Shattuck, "Combinatorial Properties of a Generalized Class of Laguerre Polynomials," Journal of Integer Sequences 24 (2021), Article 21.2.1, Theorem, equations, lines 330–339, https://cs.uwaterloo.ca/journals/JIS/VOL24/Shattuck/shattuck24.tex

Uses 1

Proof usesLaguerre–Stirling–Bell connection pair

2 fully proved

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