Newton's identities for summable sequences

TheoremNewton's identities for summable families#✓fully proved

Let \((a_j)_{j \ge 0}\) be a sequence of complex numbers, and let \((e_n)_{n \ge 0}\) and \((p_k)_{k \ge 1}\) be complex numbers such that for every \(n\) the family \(\prod_{j \in s} a_j\), indexed by the \(n\)-element finite sets \(s\) of natural numbers, is summable with sum \(e_n\), and for every \(k \ge 1\) the series \(\sum_{j \ge 0} a_j^k\) is summable with sum \(p_k\). Then for every natural number \(n\),

\[n e_n = \sum_{k=1}^{n} (-1)^{k-1} p_k e_{n-k}.\]

No absolute summability of \((a_j)\) is assumed, and the case \(n = 0\), where both sides vanish, is included.

theorem ramanujan_part1_ch3_entry11_corollary_newton_girard_general
    (a : ℕ → ℂ) (e : ℕ → ℂ) (p : ℕ → ℂ)
    (he : ∀ n : ℕ,
        HasSum (fun s : { s : Finset ℕ // s.card = n } => ∏ j ∈ (s : Finset ℕ), a j) (e n))
    (hp : ∀ k : ℕ, 1 ≤ k → HasSum (fun j : ℕ => (a j) ^ k) (p k)) :
    ∀ n : ℕ, (n : ℂ) * e n = ∑ k ∈ Finset.Icc 1 n, (-1 : ℂ) ^ (k - 1) * p k * e (n - k)

Used by 1

Used byNewton's identities for absolutely summable sequences

TheoremNewton's identities for absolutely summable sequences#✓fully proved

Let \((a_j)_{j \ge 0}\) be a sequence of complex numbers with \(\sum_j |a_j| < \infty\). Let \((e_n)_{n \ge 0}\) and \((p_k)_{k \ge 1}\) be complex numbers such that for every \(n\) the family \(\prod_{j \in s} a_j\), indexed by the \(n\)-element finite sets \(s\) of natural numbers, is summable with sum \(e_n\), and for every \(k \ge 1\) the series \(\sum_{j \ge 0} a_j^k\) is summable with sum \(p_k\). Then for every \(n \ge 1\),

\[n e_n = \sum_{k=1}^{n} (-1)^{k-1} p_k e_{n-k}.\]

theorem ramanujan_part1_ch3_entry11_corollary_newton_girard
    (a : ℕ → ℂ) (e : ℕ → ℂ) (p : ℕ → ℂ)
    (ha : Summable (fun j : ℕ => ‖a j‖))
    (he : ∀ n : ℕ,
        HasSum (fun s : { s : Finset ℕ // s.card = n } => ∏ j ∈ (s : Finset ℕ), a j) (e n))
    (hp : ∀ k : ℕ, 1 ≤ k → HasSum (fun j : ℕ => (a j) ^ k) (p k)) :
    ∀ n : ℕ, 1 ≤ n → (n : ℂ) * e n = ∑ k ∈ Finset.Icc 1 n, (-1 : ℂ) ^ (k - 1) * p k * e (n - k)

Sources

• Berndt, Ramanujan's Notebooks Part I, Chapter 3, Corollary to Entry 11, printed p. 66 / PDF p. 76.

Uses 1

Proof usesNewton's identities for summable families

TheoremPowers of an absolutely summable sequence are absolutely summable#✓fully proved

Let \((a_j)_{j \ge 0}\) be a sequence of complex numbers with \(\sum_j |a_j| < \infty\), and let \(n \ge 1\) be an integer. Then \(\sum_j |a_j^n| < \infty\).

theorem ramanujan_part1_ch3_entry11_corollary_power_summable (a : ℕ → ℂ)
    (ha : Summable (fun j => ‖a j‖)) (n : ℕ) (hn : 1 ≤ n) :
    Summable (fun j => ‖(a j) ^ n‖)

Sources

• Berndt, Ramanujan's Notebooks Part I, Chapter 3, Corollary to Entry 11, printed p. 66 / PDF p. 76.

3 fully proved

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