Power series and series transformations

Identities and transformations of power series and exponential-type series: Lagrange inversion and the tree function, Abel- and Jensen-type exponential series, coefficients of exp of a power series and Newton's identities for summable sequences, transformations of factorial series and Taylor-shift transformations, and Touchard / generalized Bell series.

• Abel-type exponential series
• Lagrange inversion series
• Newton's identities for summable sequences
• Series transformations via Taylor shifts
• Touchard polynomials and generalized Bell series
• Transformations of factorial series

TheoremRecurrence for the coefficients of \(e^{f(x)}\)#✓fully proved

Let \((A_k)_{k \ge 1}\) and \((p_n)_{n \ge 0}\) be sequences of real numbers, let \(R > 0\), and let \(f : \mathbb{R} \to \mathbb{R}\). Suppose that for every real \(x\) with \(|x| < R\) the series \(\sum_{k \ge 1} A_k x^k / k\) is summable with sum \(f(x)\) and the series \(\sum_{n \ge 0} p_n x^n\) is summable with sum \(e^{f(x)}\). Then for every integer \(n \ge 1\),

\[n p_n = \sum_{k=1}^{n} A_k p_{n-k}.\]

theorem ramanujan_part1_ch3_entry11 (A p : ℕ → ℝ) (R : ℝ) (f : ℝ → ℝ) (hR : 0 < R)
    (hf : ∀ x : ℝ, |x| < R → HasSum (fun n : ℕ => A (n + 1) * x ^ (n + 1) / (↑(n + 1) : ℝ)) (f x))
    (hg : ∀ x : ℝ, |x| < R → HasSum (fun n : ℕ => p n * x ^ n) (Real.exp (f x))) :
    ∀ n : ℕ, 0 < n → (↑n : ℝ) * p n = ∑ k ∈ Finset.Icc 1 n, A k * p (n - k)

Sources

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

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