Loading

An Index-Jets Framework for Irrationality: from Sturm-Liouville Operators to the AGM
K. Srinivasa Raghava1, R. Sivaraman2

1K. Srinivasa Raghava, Research Associate, Department of Mathematics, Choolaimedu, Chennai (Tamil Nadu), India.

2Dr. R. Sivaraman, Associate Professor, Department of Mathematics, Dwaraka Doss Goverdhan Doss Vaishnav College, Arumbakkam, Chennai (Tamil Nadu), India.Β 

Manuscript received on 29 September 2025 | Revised Manuscript received on 04 October 2025 | Manuscript Accepted on 15 October 2025 | Manuscript published on 30 October 2025Β | PP: 70-73Β | Volume-5 Issue-2, October 2025 | Retrieval Number: 100.1/ijam.B122405021025 | DOI: 10.54105/ijam.B1224.05021025

Open Access | Editorial and Publishing Policies | Cite | Zenodo | OJS | Indexing and Abstracting
Β© The Authors. Published by Lattice Science Publication (LSP). This is an open-access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Abstract: We describe a general method that proves irrationality statements from second-order equations and from the arithmetic geometric mean (AGM). Let 𝑳 > 𝟎 and define the bump 𝑩𝒏,𝑳 (𝒙) = 𝑸𝒏 𝒏! [𝒙(𝑳 βˆ’ 𝒙)]𝒏 (𝟎 ≀ 𝒙 ≀ 𝑳), where 𝑳 = 𝑷/𝑸 ∈ β„š is in lowest terms. If 𝒖 solves a second-order equation on [𝟎,𝑳] and its PrΓΌfer phase turns by an integer multiple of 𝝅, then repeated integration by parts shows that 𝑰𝒏: = βˆ«π‘³πŸŽ 𝑩𝒏,𝑳 (𝒙)𝒖(𝒙)𝒅𝒙 is an integer combination of endpoint jets and of a fixed index term. Hence 𝑰𝒏 ∈ β„€ (or 𝑫𝒏𝑰𝒏 ∈ β„€ 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 ≀ π‘³πŸπ’+𝟏 𝑸𝒏𝒏!/ (πŸπ’+𝟏)! βŸΆπ’β†’βˆž 𝟎 so for large 𝒏 we have 𝟎 < 𝑰𝒏 < 𝟏, which contradicts integrality. This scheme yields: (i) the irrationality of 𝝅 from 𝒖(𝒙) = π’”π’Šπ’ 𝒙 on [𝟎,𝝅]; (ii) a Sturm-Liouville version for analytic potentials with rational endpoint Taylor data and a halfturn of the PrΓΌfer 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 π’Œ ∈ (𝟎,𝟏) not all of 𝑲(π’Œ),𝑲(π’Œ β€²),𝑬(π’Œ),𝑬(π’Œ β€²) can be rational, and at π’Œ = 𝟏/√𝟐 at least one of 𝑲(π’Œ) or 𝑬(π’Œ) is irrational.

Keywords: Irrationality Statements, Coefficients, Function Estimate.
Scope of the Article: Applied Mathematics

Download PDF