Poly-Bernoulli numbers

DefinitionPoly-Bernoulli numbers#✓fully proved

Let \(S(n, r)\) denote the Stirling numbers of the second kind. For a natural number \(n\) and an integer \(k\), the poly-Bernoulli number \(B_n^{(k)}\) is the rational number given by Kaneko's explicit formula

\[B_n^{(k)} = (-1)^n \sum_{r=0}^{n} \frac{(-1)^r\, r!\, S(n, r)}{(r+1)^k}.\]

Introduced by M. Kaneko, "Poly-Bernoulli numbers," Journal de Théorie des Nombres de Bordeaux 9 (1997), 221–228; see also T. Arakawa and M. Kaneko, "On poly-Bernoulli numbers," Commentarii Mathematici Universitatis Sancti Pauli 48 (1999), 159–167, as cited by the source.

def polyBernoulli (n : ℕ) (k : ℤ) : ℚ :=
  (-1 : ℚ) ^ n * ∑ r ∈ Finset.range (n + 1),
    (-1 : ℚ) ^ r * (Nat.factorial r : ℚ) * (Nat.stirlingSecond n r : ℚ) /
      (((r + 1 : ℕ) : ℚ) ^ k)
Sources
Used by 5
Used byPoly-Bernoulli polynomials · Poly-Bernoulli polynomials at zero · Poly-Bernoulli polynomials of degree zero · Sums of Bernoulli numbers via poly-Bernoulli polynomials · Arakawa–Kaneko duality for poly-Bernoulli numbers

TheoremArakawa–Kaneko duality for poly-Bernoulli numbers#✓fully proved

Let \(S(n, r)\) denote the Stirling numbers of the second kind. For a natural number \(n\) and an integer \(k\), let \(B_n^{(k)}\) be the poly-Bernoulli number given by Kaneko's explicit formula

\[B_n^{(k)} = (-1)^n \sum_{r=0}^{n} \frac{(-1)^r\, r!\, S(n, r)}{(r+1)^k}.\]

Then for all natural numbers \(n\) and \(k\),

\[B_n^{(-k)} = B_k^{(-n)}.\]
theorem arakawa_kaneko_duality (n k : ℕ) :
    polyBernoulli n (-(k : ℤ)) = polyBernoulli k (-(n : ℤ))
Sources
  • Y. Hamahata and H. Masubuchi, "Special Multi-Poly-Bernoulli Numbers," Journal of Integer Sequences 10 (2007), Article 07.4.1, https://cs.uwaterloo.ca/journals/JIS/VOL10/Hamahata/hamahata3.tex. The symmetric closed formula and the duality are lines 156–174 of the official TeX, attributed there to T. Arakawa and M. Kaneko, "On poly-Bernoulli numbers," Commentarii Mathematici Universitatis Sancti Pauli 48 (1999), 159–167.
Uses 1
Statement usesPoly-Bernoulli numbers

TheoremGenerating function of negative-index poly-Bernoulli numbers#✓fully proved

For natural numbers \(n, k\) let \(B_{n,k} = \sum_{j=0}^{\min(n,k)} (j!)^2 \, S(n+1, j+1) \, S(k+1, j+1)\), where \(S\) denotes the Stirling numbers of the second kind. Then for all real numbers \(x, y\) with \((e^{|x|} - 1)(e^{|y|} - 1) < 1\),

\[\sum_{n=0}^{\infty} \left( \sum_{k=0}^{\infty} B_{n,k} \frac{x^n}{n!} \frac{y^k}{k!} \right) = \frac{e^{x+y}}{e^x + e^y - e^{x+y}}.\]

By Kaneko's closed formula, \(B_{n,k}\) is the poly-Bernoulli number \(B_n^{(-k)}\) of negative index. The condition \((e^{|x|} - 1)(e^{|y|} - 1) < 1\) is the domain of absolute convergence of the double series (all \(B_{n,k} \ge 0\)). The weaker condition \(|(e^x - 1)(e^y - 1)| < 1\) does not suffice: at \(x = -3\), \(y = \log 1.9\) the outer series diverges.

Proof: The Stirling EGF is derived from the finite-difference formula, then absolute convergence permits a Fubini rearrangement and a geometric-series sum, following Hamahata–Masubuchi, Theorems 2 and 4, and Kaneko's original formula.

public theorem polyBernoulli_negative_symmetric_exponential_generating_function
    (B : ℕ → ℕ → ℝ)
    (hB : ∀ n k : ℕ,
      B n k = ∑ j ∈ Finset.range (min n k + 1),
        (j.factorial : ℝ) ^ 2 * Nat.stirlingSecond (n + 1) (j + 1) *
          Nat.stirlingSecond (k + 1) (j + 1))
    (x y : ℝ)
    (hxy : (Real.exp |x| - 1) * (Real.exp |y| - 1) < 1) :
    (∑' n : ℕ, ∑' k : ℕ,
      B n k * (x ^ n / n.factorial) * (y ^ k / k.factorial)) =
      Real.exp (x + y) /
        (Real.exp x + Real.exp y - Real.exp (x + y))
Sources
  • Y. Hamahata and H. Masubuchi, "Special Multi-Poly-Bernoulli Numbers," Journal of Integer Sequences 10 (2007), Article 07.4.1, Theorem 4 (symmetric formula), lines 180–186, https://cs.uwaterloo.ca/journals/JIS/VOL10/Hamahata/hamahata3.tex; the closed formula identifying B is Theorem 2 there, lines 156–166. Both are attributed there to M. Kaneko, "Poly-Bernoulli numbers," Journal de Théorie des Nombres de Bordeaux 9 (1997), 221–228.
Uses 1
Proof usesExponential generating function of a shifted Stirling column
3 fully proved

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