Further results on non-negativity conjectures for copartition products

Abstract

Burson and Eichhorn proposed infinite and finite coefficientwise non-negativity conjectures for products arising from copartitions. We obtain several further results. First, both products admit decompositions into elementary local blocks, and the finite conjecture reduces exactly to its diagonal specialization. The finite conjecture is proved when the first truncation parameter is one or when the two residue parameters are equal. We also prove that every finite product has non-negative coefficients below the third shifted signed exponent. For fixed finite parameters, the coefficient sequence is shown to be eventually quasipolynomial; an explicit period bound, an explicit onset for the quasipolynomial formula, and the leading term on every residue class are obtained. This yields eventual strict positivity, apart from forced odd-degree zeros in one solved diagonal case. For the infinite conjecture, we prove non-negativity under the broader condition $a\equiv b\pmod m$, which includes the known case $a=b$. Finally, for arbitrary $b\mid a$, we improve the uniform initial non-negative range from degrees below $a+m$ to degrees below $a+2m$.