Hurwitz transform of sequences

DefinitionEntries of the Hurwitz matrix#✓fully proved

For sequences \(a, b \colon \mathbb{N} \to R\) in a commutative ring \(R\) and indices \(i, j \in \mathbb{N}\) (counted from zero), the Hurwitz entry \(h_{ij}\) is \(0\) if \(i \ge 2j + 2\); otherwise it is \(a_{j - i/2}\) if \(i\) is even and \(b_{j - (i-1)/2}\) if \(i\) is odd.

def hurwitzEntry (a b : ℕ → R) (i j : ℕ) : R :=
  if 2 * j + 2 ≤ i then 0
  else if i % 2 = 0 then a (j - i / 2)
  else b (j - (i - 1) / 2)

Sources

• Barry's definition in the Journal of Integer Sequences, vol. 15.

Used by 1

Used byHurwitz matrix

DefinitionHurwitz matrix#✓fully proved

For sequences \(a, b \colon \mathbb{N} \to R\) in a commutative ring \(R\) and \(n \in \mathbb{N}\), the Hurwitz matrix of order \(n\) is the \((n+1) \times (n+1)\) matrix whose entry in row \(i\) and column \(j\), for \(0 \le i, j \le n\), is the Hurwitz entry \(h_{ij}\) of \(a\) and \(b\).

def hurwitzMatrix (a b : ℕ → R) (n : ℕ) : Matrix (Fin (n + 1)) (Fin (n + 1)) R :=
  Matrix.of fun i j => hurwitzEntry a b i.val j.val

Uses 1 · Used by 1

Statement usesEntries of the Hurwitz matrix

Used byHurwitz transform

DefinitionHurwitz transform#✓fully proved

For sequences \(a, b \colon \mathbb{N} \to R\) in a commutative ring \(R\), the Hurwitz transform of \(a\) and \(b\) is the sequence whose \(n\)-th term, for \(n \in \mathbb{N}\), is the determinant of the Hurwitz matrix of \(a\) and \(b\) of order \(n\).

def hurwitzTransform (a b : ℕ → R) (n : ℕ) : R :=
  (hurwitzMatrix a b n).det

Uses 1 · Used by 1

Statement usesHurwitz matrix

Used byHurwitz transform of a single sequence

DefinitionHurwitz transform of a single sequence#✓fully proved

For a sequence \(a \colon \mathbb{N} \to R\) in a commutative ring \(R\), its Hurwitz transform is the Hurwitz transform of \(a\) and the sequence \((0^k)_{k \in \mathbb{N}}\), which is \(1\) at \(k = 0\) and \(0\) for \(k \ge 1\).

def hurwitzSingle (a : ℕ → R) (n : ℕ) : R :=
  hurwitzTransform a (fun k => (0 : R) ^ k) n

Uses 1

Statement usesHurwitz transform

4 fully proved

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