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

TL;DR

This paper demonstrates that rearranging terms in higher-order holomorphic recurrence relations can reduce convergence radius, challenging classical assumptions and proposing a new principle to preserve analytic integrity.

Contribution

It introduces the Principle of Indivisible Integrity, an axiom ensuring the preservation of order in recurrence relations to maintain convergence properties.

Findings

Rearrangement reduces convergence radius in higher-order recurrences

Violating the principle can cause divergence despite classical convergence conditions

Numerical examples confirm the importance of order preservation in analytic solutions

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.