Core Concepts
The canonical iterates h0, h1, h2, ... converging to the unique solution g of the iterative differential equation −γg′ = g−1 are parametrized by polynomials over the rational numbers, and the corresponding constant γ = κ ≈ 0.278877 is estimated by rational numbers.
Abstract
The content discusses the polynomial parametrization of the canonical iterates h0, h1, h2, ... that converge to the unique solution g of the iterative differential equation −γg′ = g−1, where γ > 0.
Key highlights:
The iterates h0, h1, h2, ... are constructed using the operator T defined by (Tg)(x) = ∫¹_x g*(∫g), where g* is the pseudo-inverse of g.
The iterates satisfy the relation −κnh'_n+1 = h*_n, where κn = ∫hn.
The limit h = limn→∞hn is the unique fixed point of T, and κ = ∫h = −1/h'(0) ≈ 0.278877 is the only γ > 0 for which the IDE has a solution.
The compositions qn = hn ∘ ... ∘ h1 are shown to be polynomials over the rationals, with their degrees given by the Fibonacci sequence.
Observations are made about the numerators and denominators of the κn, as well as the coefficients of the primitive polynomial representations of the qn.
Bounds and conjectures are provided for the convergence rate of the sequence (κn)n∈N₀ and the value of the stribolic constant κ.
Stats
κ0 = 1
κ1 = 1/2
κ2 = 1/3
κ3 = 3/10