General arithmetical Eulerian polynomials

DefinitionGeneral arithmetical Eulerian number#✓fully proved

Let \(R\) be a ring, let \(n\) and \(k\) be natural numbers, and let \(a, d \in R\). The general arithmetical Eulerian number \(A_{n,k}(a,d) \in R\) is

\[A_{n,k}(a,d) = \sum_{i=0}^{k} (-1)^i \binom{n+1}{i} \bigl((k+1-i)\,d - a\bigr)^n.\]

def generalArithmeticalEulerianNumber (R : Type*) [Ring R] (n k : ℕ) (a d : R) : R :=
  ∑ i ∈ Finset.range (k + 1),
    (-1 : R) ^ i * (Nat.choose (n + 1) i : R) * ((k + 1 - i : ℕ) * d - a) ^ n

Used by 1

Used byGeneral arithmetical Eulerian polynomial

DefinitionGeneral arithmetical Eulerian polynomial#✓fully proved

Let \(R\) be a ring, let \(n\) be a natural number, and let \(a, d \in R\). For each natural number \(k\), write \(A_{n,k}(a,d) \in R\) for the general arithmetical Eulerian number. The general arithmetical Eulerian polynomial is the polynomial in the indeterminate \(t\)

\[P_n(t;a,d) = \sum_{k=0}^{n} A_{n,k}(a,d)\, t^k \in R[t].\]

noncomputable def generalArithmeticalEulerianPolynomial
    (R : Type*) [Ring R] (n : ℕ) (a d : R) : Polynomial R :=
  ∑ k ∈ Finset.range (n + 1),
    Polynomial.C (generalArithmeticalEulerianNumber R n k a d) * Polynomial.X ^ k

Uses 1

Statement usesGeneral arithmetical Eulerian number

2 fully proved

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