A Reduction Library for Polynomial

Computer Science > Symbolic Computation [Submitted on 22 Jun 2026] Title:A Reduction Library for Polynomial-Base Harmonic Numbers View PDF HTML (experimental)Abstract:We develop a finite reduction library for multiple polynomial-base harmonic numbers, strict colored nested sums in which the denominator letters at each summation level are univariate polynomials. The affine and ordinary finite multiple harmonic numbers appear as lower-complexity subclasses, corresponding respectively to degree-one polynomial letters and to the ordinary letter $k$. The main mechanisms include local normalization, rational single-level descent, Euclidean division, partial fractions, factorization of polynomial letters into affine letters, quadratic splitting, exact summation of empty and polynomial-numerator levels, affine shift and lattice reductions, staircase and complement transformations, repeated-level Newton reductions, weak-to-strict diagonal decompositions, and terminal ordinary harmonic-number reductions. The accompanying Mathematica package provides a compact executable reduction library for polynomial-base, affine, and ordinary finite harmonic-number objects, with many checked examples recorded in a supplementary data-mine notebook. The current supplementary rule inventory indexes roughly 670 reduction, guard, and normalization entries, of which about 160 are named family-level entries. The library is intentionally conservative: rules are applied only under explicit hypotheses, such as absence of poles on the finite summation range, integer-power assumptions for partial-fraction descent, branch-safe scaling, finite factorization over an allowed coefficient extension, and, for telescoping, a verifiable certificate. Submission history From: Jayanta Kumar Phadikar [view email][v1] Mon, 22 Jun 2026 10:06:21 UTC (370 KB) Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.