Noncentral Whitney numbers and Tanny–Dowling polynomials

DefinitionNoncentral Whitney numbers of the second kind#✓fully proved

For elements \(m\) and \(a\) of a commutative ring \(R\), the noncentral Whitney numbers of the second kind \(W_{m,a}(n, k) \in R\), for natural numbers \(n\) and \(k\), are defined by \(W_{m,a}(0, 0) = 1\), \(W_{m,a}(0, k) = 0\) for \(k \ge 1\), \(W_{m,a}(n + 1, 0) = (-a)^{n+1}\), and

\[W_{m,a}(n + 1, k + 1) = W_{m,a}(n, k) + ((k + 1) m - a) \, W_{m,a}(n, k + 1).\]

public def noncentralWhitneySecond (m a : R) : ℕ → ℕ → R
  | 0, 0 => 1
  | 0, _ + 1 => 0
  | n + 1, 0 => (-a) ^ (n + 1)
  | n + 1, k + 1 =>
    noncentralWhitneySecond m a n k +
      (m * ((k + 1 : ℕ) : R) - a) * noncentralWhitneySecond m a n (k + 1)

Sources

• https://cs.uwaterloo.ca/journals/JIS/VOL24/Mangontarum/mang13.tex, recurrence underlying the Tanny-Dowling polynomial in equation (6).

Used by 1

Used byNoncentral Tanny–Dowling polynomials

DefinitionNoncentral Tanny–Dowling polynomials#✓fully proved

For elements \(m\) and \(a\) of a commutative ring \(R\) and a natural number \(n\), the noncentral Tanny–Dowling polynomial \(D_{m,a,n}(x) \in R[x]\) is

\[D_{m,a,n}(x) = \sum_{k=0}^{n} k! \, W_{m,a}(n, k) \, x^k,\]

where \(W_{m,a}(n, k)\) are the noncentral Whitney numbers of the second kind.

public noncomputable def noncentralTannyDowlingPolynomial (m a : R) (n : ℕ) : Polynomial R :=
  Finset.sum (Finset.range (n + 1)) fun k =>
    Polynomial.C (((Nat.factorial k : ℕ) : R) * noncentralWhitneySecond m a n k) *
      Polynomial.X ^ k

Sources

• equation (6), source https://cs.uwaterloo.ca/journals/JIS/VOL24/Mangontarum/mang13.tex.

Uses 1

Statement usesNoncentral Whitney numbers of the second kind

2 fully proved

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