An Index-Jets Framework for Irrationality: from Sturm-Liouville Operators to the AGM

Abstract: We describe a general method that proves irrationality statements from second-order equations and from the arithmeticgeometric mean (AGM). Let 𝑳 > 𝟎 and define the bump 𝑩𝒏,𝑳 (𝒙) = 𝑸 𝒏 𝒏! [𝒙(𝑳 − 𝒙)] 𝒏 (𝟎 ≤ 𝒙 ≤ 𝑳), where 𝑳 = 𝑷/𝑸 ∈ ℚ is in lowest terms. If 𝒖 solves a second-order equation on [𝟎,𝑳] and its Prüfer phase turns by an integer multiple of 𝝅, then repeated integration by parts shows that 𝑰𝒏: = ∫ 𝑳 𝟎 𝑩𝒏,𝑳 (𝒙)𝒖(𝒙)𝒅𝒙 is an integer combination of endpoint jets and of a fixed index term. Hence 𝑰𝒏 ∈ ℤ (or 𝑫𝒏𝑰𝒏 ∈ ℤ for a controlled denominator 𝑫𝒏 that depends only on finitely many endpoint Taylor coefficients of the coefficients of the equation). A Beta-function estimate gives 𝟎 < 𝑰𝒏 ≤ 𝑳 𝟐𝒏+𝟏 𝑸 𝒏𝒏! (𝟐𝒏+𝟏)! ⟶𝒏→∞ 𝟎 so for large 𝒏 we have 𝟎 < 𝑰𝒏 < 𝟏, which contradicts integrality. This scheme yields: (i) the irrationality of 𝝅 from 𝒖(𝒙) = 𝒔𝒊𝒏 𝒙 on [𝟎,𝝅]; (ii) a Sturm-Liouville version for analytic potentials with rational endpoint Taylor data and a halfturn of the Prüfer phase; and (iii) consequences for complete elliptic integrals, where the role of the index is played by the Legendre monodromy identity. In particular, for each 𝒌 ∈ (𝟎,𝟏) not all of 𝑲(𝒌),𝑲(𝒌 ′ ),𝑬(𝒌),𝑬(𝒌 ′ ) can be rational, and at 𝒌 = 𝟏/√𝟐 at least one of 𝑲(𝒌) or 𝑬(𝒌) is irrational.