Moment generating functions as J-fractions

TheoremMoment generating function as a J-fraction, general form#✓fully proved

Let \(p_0, p_1, \dots\) be real polynomials, let \(L\) be an \(\mathbb{R}\)-linear functional on \(\mathbb{R}[x]\), and let \((\alpha_n)\), \((\beta_n)\) and \((\mu_k)\) be real sequences such that \(p_0 = 1\), \(p_1 = x - \alpha_0\), \(p_{n+1} = (x - \alpha_n) p_n - \beta_n p_{n-1}\) for \(n \ge 1\), \(\mu_k = L(x^k)\) for every \(k\), and \(L(p_n) = 0\) for \(n \ge 1\). Then there are formal power series \(S_0, S_1, \dots\) over \(\mathbb{R}\), each with constant term \(1\), giving \(\mu\) a Jacobi fraction with coefficients \(\alpha\) and \(\beta\), numerator \(\mu_0\) and tails \(S_n\): that is, \(\sum_{k \ge 0} \mu_k x^k = \mu_0 S_0\) and \((1 - \alpha_n x - \beta_{n+1} x^2 S_{n+1}) S_n = 1\) for every \(n \ge 0\).

No monicity, degree or \(\beta_n \ne 0\) hypothesis is needed, and orthogonality enters only through \(L(p_n) = 0\) for \(n \ge 1\).

theorem moment_generating_function_as_J_fraction_general
    (p : ℕ → Polynomial ℝ) (L : Polynomial ℝ →ₗ[ℝ] ℝ) (α β μ : ℕ → ℝ)
    (h0 : p 0 = 1) (h1 : p 1 = Polynomial.X - Polynomial.C (α 0))
    (hrec : ∀ n : ℕ, 1 ≤ n → p (n + 1) =
      (Polynomial.X - Polynomial.C (α n)) * p n - Polynomial.C (β n) * p (n - 1))
    (hμ : ∀ k : ℕ, μ k = L (Polynomial.X ^ k))
    (hLp : ∀ n : ℕ, 1 ≤ n → L (p n) = 0) :
    ∃ S : ℕ → PowerSeries ℝ,
      (∀ n : ℕ, PowerSeries.constantCoeff (S n) = 1) ∧ HasJacobiFraction μ α β (μ 0) S

Uses 1 · Used by 1

Statement usesJacobi continued fraction expansion

Used byMoment generating function as a J-fraction

TheoremMoment generating function as a J-fraction#✓fully proved

Let \(p_0, p_1, \dots\) be real polynomials, each \(p_n\) monic of degree \(n\), let \(L\) be an \(\mathbb{R}\)-linear functional on \(\mathbb{R}[x]\), and let \((\alpha_n)\), \((\beta_n)\) and \((\mu_k)\) be real sequences. Suppose that \(p_0 = 1\), \(p_1 = x - \alpha_0\) and \(p_{n+1} = (x - \alpha_n) p_n - \beta_n p_{n-1}\) for \(n \ge 1\), that \(\beta_n \ne 0\) for \(n \ge 1\), that the \(p_n\) are orthogonal for \(L\), meaning \(L(p_m p_n) = 0\) for \(m \ne n\) and \(L(p_n^2) \ne 0\) for every \(n\), that \(\mu_k = L(x^k)\) for every \(k\), and that \(\mu_0 \ne 0\). Let \(g = \sum_{k \ge 0} \mu_k x^k\) be the moment generating function. Then there are formal power series \(S_0, S_1, \dots\) over \(\mathbb{R}\), each with constant term \(1\), such that \(g = \mu_0 S_0\) and \(S_n (1 - \alpha_n x - \beta_{n+1} x^2 S_{n+1}) = 1\) for every \(n \ge 0\).

These equations present \(g\) as the J-fraction \(\mu_0 / (1 - \alpha_0 x - \beta_1 x^2 / (1 - \alpha_1 x - \beta_2 x^2 / \cdots))\) with tails \(S_n\). The hypotheses that each \(p_n\) is monic of degree \(n\), that \(\beta_n \ne 0\), that \(L(p_n^2) \ne 0\) and that \(\mu_0 \ne 0\) are not needed, and orthogonality is used only through \(L(p_n) = 0\) for \(n \ge 1\).

The source's recurrence line reads \(\beta_n(x)\); it is read as \(\beta_n p_{n-1}\), as in the surrounding lines 315–317 and 364–366 of the source.

MetaMathlibExt.moment_generating_function_as_J_fraction · 23 lines

theorem moment_generating_function_as_J_fraction
    (p : ℕ → Polynomial ℝ)
    (L : Polynomial ℝ →ₗ[ℝ] ℝ)
    (α β : ℕ → ℝ)
    (μ : ℕ → ℝ)
    (g : PowerSeries ℝ) :
    (∀ n : ℕ, (p n).Monic ∧ (p n).natDegree = n) →
    (p 0 = 1) →
    (p 1 = Polynomial.X - Polynomial.C (α 0)) →
    (∀ n : ℕ, 1 ≤ n → p (n + 1) = (Polynomial.X - Polynomial.C (α n)) * p n - Polynomial.C (β n) * p
      (n - 1)) →
    (∀ n : ℕ, 1 ≤ n → β n ≠ 0) →
    (∀ m n : ℕ, m ≠ n → L (p m * p n) = 0) →
    (∀ n : ℕ, L (p n * p n) ≠ 0) →
    (∀ k : ℕ, μ k = L (Polynomial.X ^ k)) →
    (μ 0 ≠ 0) →
    (g = PowerSeries.mk μ) →
    ∃ S : ℕ → PowerSeries ℝ,
      (∀ n : ℕ, PowerSeries.constantCoeff (S n) = 1) ∧
      (g = PowerSeries.C (μ 0) * S 0) ∧
      ∀ n : ℕ, S n *
        (1 - PowerSeries.C (α n) * PowerSeries.X - PowerSeries.C (β (n + 1)) * PowerSeries.X ^ 2 * S
        (n + 1)) = 1

Sources

• Paul Barry and Aoife Hennessy, "Meixner-Type Results for Riordan Arrays and Associated Integer Sequences," Journal of Integer Sequences 13 (2010), Article 10.9.4, J-fraction theorem citing C. Krattenthaler (cf. Viennot Proposition 1, (7), p. V-5; Wall Theorem 51.1), lines 353–362, https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry5/barry96s.tex

Uses 2

Proof usesJacobi continued fraction expansion · Moment generating function as a J-fraction, general form

2 fully proved

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