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On the Incompatibility of Rearrangement with Convergence: An Axiomatic Approach to Holomorphic Recurrence Relations

Yoon-Seok Choun

TL;DR

The work addresses the assumption that internal rearrangement of terms does not affect convergence radii for power-series solutions, showing this fails for holomorphic solutions of higher-order recurrence relations. By introducing the Principle of Indivisible Integrity, the authors axiomatize structural constraints that preserve convergence and, crucially, uniqueness, demonstrated via Heun-type and hypergeometric recurrences. They reveal that the radius of absolute convergence under rearrangement can be strictly smaller than the classical Poincaré–Perron bound, necessitating a no-rearrangement constraint to align with the uniqueness theorem and integral-series interchange. The findings have broad implications for analytic methods in mathematical physics and automated symbolic computation, suggesting a refined framework for recurrence-based analytic techniques. In short, preserving the indivisible structure of recurrence terms yields a more rigorous, universally applicable convergence theory for higher-order systems.

Abstract

In classical analysis, the convergence behavior of power series solutions to differential or recurrence equations is generally assumed to be invariant under internal rearrangement. This paper challenges that belief by proving that, for holomorphic solutions to higher-order recurrence relations (order 3 or more), rearrangement of internal terms systematically reduces the radius of convergence. This contradicts assumptions underlying both Fuchs' theorem and the Poincare-Perron theorem. To address this, the paper proposes the Principle of Indivisible Integrity, an axiom that restricts arbitrary reordering within analytic computations. Both analytic arguments and numerical examples (see Theorem 3.3 and Table 3) show that violation of this principle can lead to structural divergence, even when classical conditions suggest convergence. This framework suggests the need to reexamine analytic structures in recurrence-based methods across mathematical physics, including quantum mechanics, general relativity, and spectral theory. It also raises foundational questions about computation and mathematical rigor in an age of automated symbolic processing. Rather than offering just a technical correction, this paper advocates a philosophical principle: that the integrity of mathematical order must be preserved by structure, not merely by computational convenience.

On the Incompatibility of Rearrangement with Convergence: An Axiomatic Approach to Holomorphic Recurrence Relations

TL;DR

The work addresses the assumption that internal rearrangement of terms does not affect convergence radii for power-series solutions, showing this fails for holomorphic solutions of higher-order recurrence relations. By introducing the Principle of Indivisible Integrity, the authors axiomatize structural constraints that preserve convergence and, crucially, uniqueness, demonstrated via Heun-type and hypergeometric recurrences. They reveal that the radius of absolute convergence under rearrangement can be strictly smaller than the classical Poincaré–Perron bound, necessitating a no-rearrangement constraint to align with the uniqueness theorem and integral-series interchange. The findings have broad implications for analytic methods in mathematical physics and automated symbolic computation, suggesting a refined framework for recurrence-based analytic techniques. In short, preserving the indivisible structure of recurrence terms yields a more rigorous, universally applicable convergence theory for higher-order systems.

Abstract

In classical analysis, the convergence behavior of power series solutions to differential or recurrence equations is generally assumed to be invariant under internal rearrangement. This paper challenges that belief by proving that, for holomorphic solutions to higher-order recurrence relations (order 3 or more), rearrangement of internal terms systematically reduces the radius of convergence. This contradicts assumptions underlying both Fuchs' theorem and the Poincare-Perron theorem. To address this, the paper proposes the Principle of Indivisible Integrity, an axiom that restricts arbitrary reordering within analytic computations. Both analytic arguments and numerical examples (see Theorem 3.3 and Table 3) show that violation of this principle can lead to structural divergence, even when classical conditions suggest convergence. This framework suggests the need to reexamine analytic structures in recurrence-based methods across mathematical physics, including quantum mechanics, general relativity, and spectral theory. It also raises foundational questions about computation and mathematical rigor in an age of automated symbolic processing. Rather than offering just a technical correction, this paper advocates a philosophical principle: that the integrity of mathematical order must be preserved by structure, not merely by computational convenience.

Paper Structure

This paper contains 44 sections, 5 theorems, 133 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Proof of the Breakdown of Absolute Convergence in Higher-Order Recurrence Relations
  4. Application to the Heun's Differential Equation
  5. P--P Theorem and Its Applications for Solutions of Power Series
  6. Structural Modifications and Radius of Convergence in a Power Series with a Three-Term Recurrence Relation
  7. Numerical Comparison of Absolute Convergence and Internal Component Rearrangement
  8. Absolute Convergence and Internal Component Rearrangement
  9. absolute convergence:
  10. Discrepancy with internal component rearrangement:
  11. Findings:
  12. Analysis of Hypergeometric Series and Radius of Convergence
  13. Asymptotic Expansion:
  14. Structural Constraints in the P--P Theorem:
  15. Special Case $a = -1$:
  16. ...and 29 more sections

Key Result

Theorem 3.2

To ensure the coefficients stabilize asymptotically, we assume $\lim_{n\rightarrow \infty}\alpha_{l,n}= \alpha_l<\infty$ in (jj:3), the minimal domain under internal component rearrangement for (jj:2) is given by: where $\mathcal{D}$ represents the minimal domain under internal component rearrangement.

Figures (5)

  • Figure 1: Domain of convergence according to the original P--P theorem.
  • Figure 2: Structural Constraints in the P--P Theorem.
  • Figure 3: Domains of convergence for absolute and the radius of convergence under internal component rearrangement.
  • Figure 4: Partial sum
  • Figure 5: Integral

Theorems & Definitions (7)

  • proof
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Theorem 4.1: P--P Theorem Miln1933
  • Corollary 7.1
  • Corollary 7.2