A Reduction Library for Polynomial-Base Harmonic Numbers

RESUMO

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.