Stieltjes production matrices

DefinitionTridiagonal Stieltjes production matrix#✓fully proved

Let \(R\) be an additive monoid with one and let \(a_1, b_1, a, b \in R\). The Stieltjes matrix \(S\) is the infinite matrix with rows and columns indexed by \(\mathbb{N} = \{0, 1, 2, \dots\}\) whose entries are \(S_{i, i+1} = 1\) for every \(i\), \(S_{0,0} = a_1\), \(S_{1,0} = b_1\), \(S_{i,i} = a\) and \(S_{i+1,i} = b\) for every \(i \ge 1\), and \(S_{i,j} = 0\) for all other \(i, j\). So row \(0\) is \(a_1, 1, 0, \dots\), row \(1\) is \(b_1, a, 1, 0, \dots\), and each row \(i \ge 2\) has \(b\) below the diagonal, \(a\) on the diagonal, \(1\) above the diagonal and \(0\) elsewhere.

def stieltjesMatrix {R : Type*} [AddMonoidWithOne R] (a1 b1 a b : R) (i j : ℕ) : R :=
  if j = i + 1 then 1
  else if i = 0 ∧ j = 0 then a1
  else if i = 1 ∧ j = 0 then b1
  else if j = i then a
  else if j + 1 = i then b
  else 0

Used by 2

Used byMeixner-type characterization of Riordan orthogonal polynomial arrays · Specialized Stieltjes matrix

DefinitionSpecialized Stieltjes matrix#✓fully proved

Let \(R\) be an additive monoid with one and let \(a, b, \lambda, \mu \in R\). The specialized Stieltjes matrix is the tridiagonal Stieltjes production matrix with \(a_1 = a + \lambda\), \(b_1 = b + \mu\) and the same \(a\) and \(b\): the infinite matrix \(S\) with rows and columns indexed by \(\mathbb{N} = \{0, 1, 2, \dots\}\) whose entries are \(S_{i, i+1} = 1\) for every \(i\), \(S_{0,0} = a + \lambda\), \(S_{1,0} = b + \mu\), \(S_{i,i} = a\) and \(S_{i+1,i} = b\) for every \(i \ge 1\), and \(S_{i,j} = 0\) for all other \(i, j\).

This specialization corresponds to the inverse Riordan pair with numerator \(1 - \lambda x - \mu x^2\) over \(1 + a x + b x^2\).

def stieltjesMatrixOf {R : Type*} [AddMonoidWithOne R] (a b lam mu : R) (i j : ℕ) : R :=
  stieltjesMatrix (a + lam) (b + mu) a b i j

Uses 1

Statement usesTridiagonal Stieltjes production matrix

2 fully proved

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