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

Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 70 Retrieval Number:100.1/ijam.B122405021025 DOI: 10.54105/ijam.B1224.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. An Index-Jets Framework for Irrationality: from Sturm-Liouville Operators to the AGM K. Srinivasa Raghava, R. Sivaraman 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)= 𝑸𝒏 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 𝟎𝑩𝒏,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≤ 𝑳𝟐𝒏+𝟏 𝑸𝒏𝒏! (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. Keywords: Irrationality Statements, Coefficients, Function Estimate I. INTRODUCTION Purpose. This paper gives a single method that proves irrationality statements by combining two ideas: (i) a bump with high-order zeros at the endpoints whose derivatives ("jets") become integers if the interval length is rational, and (ii) an index that is inherently an integer, coming from a phase turn for second-order equations or from a monodromy identity for complete elliptic integrals. Objectives. We aim to: ▪Present the index-jets mechanism in a form that is easy to apply and check; ▪Recover the irrationality of 𝜋 from the constant curvature model 𝑢′′+𝑢=0 on [0,𝜋]; Manuscript received on 29 September 2025 | Revised Manuscript received on 04 October 2025 | Manuscript Accepted on 15 October 2025 | Manuscript published on 30 October 2025. *Correspondence Author(s) K. Srinivasa Raghava, Research Associate, Department of Mathematics, Choolaimedu, Chennai (Tamil Nadu), India. Email ID: [email protected], ORCID ID: 0000-0003-0021-6322 Dr. R. Sivaraman*, Associate Professor, Department of Mathematics, Dwaraka Doss Goverdhan Doss Vaishnav College, Arumbakkam, Chennai (Tamil Nadu), India. Email ID: [email protected], ORCID ID: 0000-0001-5989-4422 © 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/ ▪Give a Sturm-Liouville version where the index is a half-turn of the Prüfer phase (see [1, Ch. 5] and [5, Sec. 11]); ▪State consequences for complete elliptic integrals using Legendre's relation (a monodromy identity of the Gauss hypergeometric equation, see [2, §22] and [4, 19.12]); ▪Keep the analytic estimate uniform via a short Betafunction bound, and point to fast AGM convergence where relevant (see [3, 6]). Background and relation to prior work. “Classical proofs that π is irrational use an integral with a polynomial weight that vanishes at the endpoints, and then integrate by parts repeatedly.” Our contribution is to package this into an index + jets template and to show that the same template works in three places: constant curvature, Sturm-Liouville form, and the AGM/elliptic setting. The integer does not come from ad hoc cancellation but from a genuine integer-valued quantity: “... a Prüfer phase turn (Sturm oscillation theory [1, Ch. 5], [5, Sec. 11]) or Legendre monodromy ...” ([2, §22], [4, 19.12]). For the AGM, we follow [3, Chs. 1-3] and use the standard “...identities for K and E together with fast convergence …” (see also [6]). “Method. If L = P/Q is rational, the scaled bump ...” 𝐵𝑛,𝐿(𝑥)=𝑄𝑛 𝑛![𝑥(𝐿−𝑥)]𝑛 has integer jets of order ≥𝑛 at 0 and at 𝐿. After 2𝑛 integrations by parts against a special solution 𝑢, the target integral equals a sum of integer boundary jets plus a single interior term that is controlled by an index (phase turn or monodromy). This makes the integral an integer. A Betafunction bound then shows the integral tends to 0 as 𝑛→∞, so for large 𝑛 it lies in (0,1), which is impossible. Scope and limitations. In the Sturm-Liouville setting, we assume analytic coefficients near the endpoints and rational endpoint Taylor data, which allows us to control denominators in the boundary terms. We do not pursue deep transcendence results; all consequences here come from the index and the Beta bound. Notation. We write 𝐿>0 for the interval length, 𝐵𝑛,𝐿 for the bump, 𝐼𝑛=∫ 𝐿 0𝐵𝑛,𝐿𝑢 for the main integral, and 𝜃 for a Prüfer angle when ℒ𝑢=𝜆𝑢 is in Sturm-Liouville form. We use 𝐾(𝑘) and 𝐸(𝑘) for complete elliptic integrals and 𝑘′= √1−𝑘2 for the complementary modulus ([4, Ch. 19], [2]). Structure of the paper. Section 2 recalls the integer jets and the Beta bound. Section 3 applies the method to 𝑢′′+𝑢=0 on [0,𝜋] to get the irrationality of 𝜋. Section 4 gives the SturmLiouville variant with the Prüfer index (citing [1, 5]). Section 5 treats the An Index-Jets Framework for Irrationality: from Sturm-Liouville Operators to the AGM 71 Retrieval Number:100.1/ijam.B122405021025 DOI: 10.54105/ijam.B1224.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. AGM/elliptic case using [3, 4, 2] and states consequences for 𝐾 and 𝐸. Section 6 contains two short calculations that make the integrality and the bound explicit. Section 7 summarizes the method and points to two extensions. II. THE BASIC TOOLS Integer jets at the endpoints. Let 𝐿>0 and 𝑛≥1. Define 𝐵𝑛,𝐿(𝑥):=𝑄𝑛 𝑛![𝑥(𝐿−𝑥)]𝑛,0≤𝑥≤𝐿 where 𝐿=𝑃/𝑄 is in lowest terms. Then 𝐵𝑛,𝐿 (𝑚)(0)= 𝐵𝑛,𝐿 (𝑚)(𝐿)=0 for 𝑚<𝑛. For 𝑚=𝑛+𝑟 with 0≤𝑟≤𝑛, 𝐵𝑛,𝐿 (𝑛+𝑟)(0)=(−1)𝑟(𝑛+𝑟)! 𝑟!(𝑛−𝑟)!𝑃𝑛−𝑟𝑄𝑟∈ℤ,𝐵𝑛,𝐿 (𝑛+𝑟)(𝐿) =(−1)𝑛+𝑟𝐵𝑛,𝐿 (𝑛+𝑟)(0) A clean smallness bound. The Beta-function identity gives ∫ 𝐿 0[𝑥(𝐿−𝑥)]𝑛𝑑𝑥=𝐿2𝑛+1∫ 1 0[𝑡(1−𝑡)]𝑛𝑑𝑡 =𝐿2𝑛+1 (𝑛!)2 (2𝑛+1)! If 0≤𝑤≤1 on [0,𝐿], then ∫ 𝐿 0𝐵𝑛,𝐿(𝑥)𝑤(𝑥)𝑑𝑥≤𝐿2𝑛+1 𝑄𝑛𝑛! (2𝑛+1)! ⟶ 𝑛→∞0 Index, degree, or monodromy. Three sources of integers appear here. ▪ For 𝑢′′+𝜅𝑢=0, the Prüfer angle 𝜃 rotates by √𝜅𝐿. If 𝑢(0)=𝑢(𝐿)=0 and 𝑢≢0, then 𝜃(𝐿)−𝜃(0)= 𝑚𝜋 with 𝑚∈ℤ (Sturm oscillation; see [1, Ch. 5]). ▪ For the complete elliptic integrals 𝐾(𝑘) and 𝐸(𝑘), Legendre's relation 𝐾(𝑘)𝐸(𝑘′)+𝐾(𝑘′)𝐸(𝑘)−𝐾(𝑘)𝐾(𝑘′)=𝜋 2,𝑘′=√1−𝑘2 is a monodromy invariant of the hypergeometric equation [2, 4]. ▪ On [0,𝜋/2], the identity ∫ 𝜋/2 0sin (2𝜃)𝑑𝜃=1 is a basic degree count. III. CONSTANT CURVATURE MODEL: IRRATIONALITY OF 𝝅 Take 𝜅=1 and 𝑢(𝑥)=sin 𝑥 on [0,𝜋]. Define 𝐼𝑛:=∫ 𝜋 0𝐵𝑛,𝜋(𝑥)sin 𝑥𝑑𝑥,𝐵𝑛,𝜋(𝑥)=𝑞𝑛 𝑛![𝑥(𝜋−𝑥)]𝑛 Assume for contradiction that 𝜋=𝑝/𝑞 in lowest terms. Integrate 𝐼𝑛 by parts 2𝑛 times. All low-order boundary terms vanish. The remaining boundary terms are integers by the jet formula. The interior remainder contains 𝐵𝑛,𝜋 (2𝑛), which is constant and equal to 𝑞𝑛(2𝑛)! 𝑛! up to sign. The sign from 2𝑛 integrations cancel the leading sign in [𝑥(𝜋−𝑥)]𝑛, and ∫ 𝜋 0sin 𝑥𝑑𝑥=2 so 𝐼𝑛∈ℤ. The Beta bound gives 𝐼𝑛→0, hence for large 𝑛 we have 0<𝐼𝑛<1, a contradiction. Therefore 𝜋 is irrational. IV. A STURM-LIOUVILLE VERSION Let ℒ𝑦:=−𝑦′′+ 𝑉(𝑥)𝑦 on [0,𝐿], with 𝑉 real analytic near [0,𝐿] and with rational Taylor coefficients at 0 and 𝐿. Suppose 𝑢 solves ℒ𝑢=𝜆𝑢, satisfies 𝑢(0)=𝑢(𝐿)=0, and its Prüfer angle turns by exactly 𝜋 on [0,𝐿]. Assume 𝐿=𝑃/𝑄∈ℚ. Set 𝐼𝑛:=∫ 𝐿 0𝐵𝑛,𝐿(𝑥)𝑢(𝑥)𝑑𝑥 Formal self-adjointness gives ∫ 𝐿 0(𝐵𝑛,𝐿ℒ𝑢−𝑢ℒ𝐵𝑛,𝐿)𝑑𝑥=[𝐵𝑛,𝐿𝑢′−𝐵𝑛,𝐿 ′𝑢]0 𝐿 Iterating moves derivatives onto 𝐵𝑛,𝐿. Low-order boundary terms vanish; higher-order boundary terms depend only on finitely many endpoint jets of 𝐵𝑛,𝐿 and 𝑉. Because the jets of 𝐵𝑛,𝐿 are integers and the jets of 𝑉 are rational, there is a common denominator 𝐷𝑛 such that 𝐷𝑛 times each boundary term is an integer. The interior term reduces to a fixed rational multiple of the average of cos 𝜃 over a half turn. Hence 𝐷𝑛𝐼𝑛∈ℤ. The Beta bound shows 𝐼𝑛→0. If the endpoint Taylor coefficients of 𝑉 are integers, we may take 𝐷𝑛=1, which yields a contradiction for large 𝑛. V. AGM AND LEGENDRE: CONSEQUENCES FOR 𝑲 AND 𝑬 Let 𝑘∈(0,1) and 𝑘′=√1−𝑘2. Gauss's identity relates 𝐾 to the AGM: 𝐾(𝑘)= 𝜋 2𝑀(1,𝑘′) [3, Chs. 1-3], [4, Ch. 19]. There is a companion AGM series for 𝐸 : 𝐸(𝑘)= 𝜋 2𝑀(1,𝑘′)(1−∑ 𝑛≥0 2𝑛−1𝑐𝑛 2). Legendre's relation is 𝐾(𝑘)𝐸(𝑘′)+𝐾(𝑘′)𝐸(𝑘)−𝐾(𝑘)𝐾(𝑘′)=𝜋 2 [2, 4]. To generate an integer on [0,𝜋/2], fix 𝑁≥1 and define Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 72 Retrieval Number:100.1/ijam.B122405021025 DOI: 10.54105/ijam.B1224.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Φ𝑁(𝜃):=(2𝑄)𝑁 𝑁! [𝜃(𝜋 2−𝜃)]𝑁,𝐽𝑁: =∫ 𝜋/2 0Φ𝑁(𝜃)sin (2𝜃)𝑑𝜃 where 𝜋/2=𝑃/(2𝑄) in lowest terms. Then 𝐽𝑁∈ℤ by the same jet argument as before, and 𝐽𝑁→0 by the Beta bound. Writing Legendre's relation as a boundary evaluation of a conserved Wronskian and “...repeating the 2N integrations by parts yields integers αN, βN, γN with ...” 𝛼𝑁𝐾(𝑘)𝐸(𝑘′)+𝛽𝑁𝐾(𝑘′)𝐸(𝑘)−𝛾𝑁𝐾(𝑘)𝐾(𝑘′)=𝐽𝑁 2 Letting 𝑁→∞ gives the following. For each 𝑘∈(0,1), not all of 𝐾(𝑘),𝐾(𝑘′),𝐸(𝑘),𝐸(𝑘′) can be rational. At 𝑘=1/√2, at least one of 𝐾(𝑘) or 𝐸(𝑘) is irrational. VI. TWO SHORT CALCULATIONS Integrality in the constant curvature model. For 𝐼𝑛=∫ 𝜋 0𝐵𝑛,𝜋(𝑥)sin 𝑥𝑑𝑥 apply integration by parts 2𝑛 times. The low-order boundary terms vanish. The high-order boundary evaluations are integer jets. The remaining interior term is 𝑞𝑛(2𝑛)! 𝑛! (−1)𝑛∫ 𝜋 0sin 𝑥𝑑𝑥 and the factor (−1)𝑛 cancels the sign produced by the 2𝑛 integrations by parts. Since ∫ 𝜋 0sin 𝑥𝑑𝑥=2, we obtain 𝑞𝑛(2𝑛)! 𝑛! ∫ 𝜋 0sin 𝑥𝑑𝑥=2𝑞𝑛(2𝑛)! 𝑛! ∈ℤ Hence 𝐼𝑛∈ℤ. The Beta bound in one line. Because 0≤sin 𝑥≤1 on [0,𝜋], 0<𝐼𝑛≤𝑞𝑛 𝑛!∫ 𝜋 0[𝑥(𝜋−𝑥)]𝑛𝑑𝑥=𝜋2𝑛+1 𝑞𝑛𝑛! (2𝑛+1)! ⟶ 𝑛→∞0 VII. CONCLUSION We gave one method for proving irrationality that combines two ideas: endpoint jets and an index. “If the interval length L = P/Q is rational (lowest terms), the scaled bump 𝐵𝑛,𝐿(𝑥)= 𝑄𝑛 𝑛![𝑥(𝐿−𝑥)]𝑛 has integer derivatives of order ≥ n at both endpoints.” After 2𝑛 integrations by parts against a suitable solution 𝑢, the integral ∫ 𝐿 0𝐵𝑛,𝐿𝑢 becomes a sum of these integer jets plus a single term fixed by an index: a half turn of a Prüfer angle in the constant curvature and Sturm-Liouville settings (see [1, 5]) or the Legendre monodromy constant for complete elliptic integrals (see [2, 4]). This makes the whole integral an integer. A short Beta-function bound shows that the same integral tends to 0 as 𝑛 grows, so for large 𝑛 it lies in (0,1), which is impossible. This recovers the irrationality of 𝜋, gives a Sturm-Liouville version under mild endpoint assumptions, and yields basic consequences for 𝐾 and 𝐸. Two natural next steps are to relax the endpoint conditions (for example, via a Liouville transform or a direct Prüfer equation) and to apply the same index-jet idea to other hypergeometric equations with known monodromy identities. DECLARATION STATEMENT Some of the references cited are older, noted explicitly as [1], [2], [3], [4], [5] and [6]. However, these works remain significant for the current study, as they are pioneering in their fields. After aggregating input from all authors, I must verify the accuracy of the following information as the article's author. ▪ Conflicts of Interest/ Competing Interests: Based on my understanding, this article has no conflicts of interest. ▪ Funding Support: This article has not been funded by any organizations or agencies. This independence ensures that the research is conducted with objectivity and without any external influence. ▪ Ethical Approval and Consent to Participate: The content of this article does not necessitate ethical approval or consent to participate with supporting documentation. ▪ Data Access Statement and Material Availability: The adequate resources of this article are publicly accessible. ▪ Author’s Contributions: The authorship of this article is contributed equally to all participating individuals. REFERENCES 1. M. P. do Carmo, Riemannian Geometry, Birkhäuser, 1992. URL: link.springer.com/book/10.1007/978-1-4757-2184-1., works remain significant, see declaration 2. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed., Cambridge Univ. Press, 1927. DOI: https://doi.org/10.1017/CBO9780511608759., works remain significant, see declaration 3. J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Wiley, 1987. DOI (eBook): DOI: https://doi.org/10.1002/9781118032576., works remain significant, see declaration 4. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark (eds.), NIST Digital Library of Mathematical Functions. URL: https://dlmf.nist.gov/19 (Elliptic integrals and AGM)., works remain significant, see declaration 5. G. Teschl, Ordinary Differential Equations and Dynamical Systems, Amer. Math. Soc., 2012. URL: https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ode.pdf. (For Prüfer transformation and Sturm oscillation; see Sec. 11.), works remain significant, see declaration 6. R. P. Brent, "Fast multiple-precision evaluation of elementary functions," J. ACM 23 (1976), 242-251. DOI: https://doi.org/10.1145/321941.321944. (For quadratic convergence underlying the AGM.), works remain significant, see declaration AUTHOR’S PROFILE K. Srinivasa Raghava is a mathematics researcher in number theory, cryptography, graph theory, and work inspired by Srinivasa Ramanujan. A mathematics and Vedic Math teacher and a chess player, he is a member of the MAA and AIRMC. He received the Prathibha Shiromani Award, the Math Genius Award, and the A.P.J. Abdul Kalam Award. He has presented papers and delivered invited talks at 72 national and 65 international conferences. His passion was to create new and exciting formulas in mathematics and extend mathematical concepts with philosophy, quantum mechanics and computer science. His posts about mathematics on various social An Index-Jets Framework for Irrationality: from Sturm-Liouville Operators to the AGM 73 Retrieval Number:100.1/ijam.B122405021025 DOI: 10.54105/ijam.B1224.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. media platforms have influenced many young students and mathematics enthusiasts throughout the globe. Dr. R. Sivaraman, working as an Associate Professor at Dwaraka Doss Goverdhan Doss Vaishnav College, Chennai, has 26 years of teaching experience at the College level. He was conferred the National Award for Popularising Mathematics among the masses in 2016 by the Department of Science and Technology, Government of India. He was conferred with the Indian National Science Academy (INSA) Teaching Award for the year 2018. He has also received State Government Best Science Book Awards in 2011 and 2012. He has provided more than 400 lectures conveying the beauty and applications of Mathematics. He has published more than 300 research papers and has done his Post Doctoral Research Fellowship and Doctor of Science Degree. He has written 47 books in view of popularizing mathematics among common man. He was a member of the Textbook Writing Committee of the Tamil Nadu School Education Department, responsible for preparing a revised mathematics textbook for the eleventh class. He served as chairperson for the tenth class. He has won more than 75 prestigious awards for his distinguished service to mathematics. He has been offering free courses to college students from impoverished backgrounds for many years. Propagating the beauty and applications of mathematics to everyone was his life mission. Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of the Lattice Science Publication (LSP)/ journal and/ or the editor(s). The Lattice Science Publication (LSP)/ journal and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.