Deferred Cyclotomic Representation for Stable and Exact Evaluation of q-Hypergeometric Series

TL;DR

The paper introduces the deferred cyclotomic representation (DCR), a novel method for exact and stable evaluation of q-hypergeometric series that reduces computational complexity and enhances numerical reliability.

Contribution

It presents the DCR framework that separates algebraic structure from evaluation, enabling exact cancellations, linear memory scaling, and a unified view of q-deformed amplitudes.

Findings

Achieves linear memory scaling in the compilation phase.

Eliminates intermediate expression swell in exact arithmetic.

Extends reliable double-precision computation range.

Abstract

We introduce a cyclotomic representation for finite $q$-hypergeometric series and $q$-deformed amplitudes that separates algebraic structure from evaluation. By expressing each summand in a sparse exponent basis over irreducible cyclotomic polynomials, all products and ratios of quantum factorials reduce to integer vector arithmetic. This ensures that cancellations between numerator and denominator are resolved exactly prior to any evaluation. This formulation yields the deferred cyclotomic representation (DCR), a parameter-independent combinatorial object of the series, from which evaluation in any target field is realized as a ring homomorphism. For quantum recoupling coefficients, we demonstrate that this framework achieves linear memory scaling in the compilation phase, eliminates intermediate expression swell in exact arithmetic, and substantially extends the range of reliable double-precision computation by reducing cancellation-induced error amplification. Beyond its computational advantages, the DCR provides a unified perspective on $q$-deformed amplitudes. Structural properties like admissibility at roots of unity, and the classical limit all emerge as intrinsic properties of a single underlying combinatorial object.