Deferred Cyclotomic Representation for Stable and Exact Evaluation of q-Hypergeometric Series
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| Format: | Preprint |
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2026
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| _version_ | 1866910134421159936 |
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| author | Asante, Seth K. |
| author_facet | Asante, Seth K. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_13196 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Deferred Cyclotomic Representation for Stable and Exact Evaluation of q-Hypergeometric Series Asante, Seth K. Mathematical Physics Numerical Analysis General Relativity and Quantum Cosmology High Energy Physics - Theory Computational Physics 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. |
| title | Deferred Cyclotomic Representation for Stable and Exact Evaluation of q-Hypergeometric Series |
| topic | Mathematical Physics Numerical Analysis General Relativity and Quantum Cosmology High Energy Physics - Theory Computational Physics |
| url | https://arxiv.org/abs/2604.13196 |