bibkey: andrews2025positive authors: George E. Andrews, Mohamed El Bachraoui year: 2025 title: “Certain positive q-series and inequalities for two-color partitions” doi: 10.48550/arXiv.2507.09276 url: https://arxiv.org/abs/2507.09276v1 claim: “Positivity questions for two families of q-series C’(k,m,n) and D’(k,m,n) generating weighted two-color partitions; positivity proofs for small parameters and Conjectures 1–5 on positivity, sign patterns and eventual positivity.” strata_touched:

• D5/S3/Combinatorics/TwoColorPartition/AndrewsElBachraouiThree
• D5/S3/Combinatorics/TwoColorPartition/AndrewsElBachraouiFour
• D5/S3/Combinatorics/TwoColorPartition/AndrewsElBachraouiTwo license: citation-only triage: anchor

Andrews and El Bachraoui, positive q-series and two-color partitions

The paper studies Σ_n C’(k,m,n) qⁿ = Σ_{j≥0} q^{m(2j+1)} (q^{2j+2}, q^{2j+2k}; q²)∞ / (q^{2j+1}; q²)∞² and Σ_n D’(k,m,n) qⁿ = Σ_{j≥0} q^{m(2j+2)} (q^{2j+4}, q^{2j+2+2k}; q²)∞ / (q^{2j+3}; q²)∞², whose coefficients are differences D₀ − D₁ of two-color partitions counted by the parity of the number of even parts. Theorem 1 transforms C’ into a single sum with finite products, and Theorem 4 relates D’ to C’. The paper proves positivity of C’(k,1) for k ≤ 4 and of D’(2,1), and conjectures positivity of C’(k,1) for all k (Conjecture 1), of C’(2,4) (Conjecture 2) and of D’(2,2) (Conjecture 3), that D’(2,3,n) < 0 exactly for n = 10, 22 (Conjecture 4), and eventual positivity of D’(k,m) for k > m (Conjecture 5). Conjecture 1 is proved by Chern and Wang, arXiv:2601.19845.

The module D5/S3/Combinatorics/TwoColorPartition/AndrewsElBachraouiThree proves Conjecture 3.

Verified locator

DOI: 10.48550/arXiv.2507.09276

URL: https://arxiv.org/abs/2507.09276v1

• Locator: Section 1, equations (1.2) and (1.3).
• Locator: Section 2, Theorem 1 and Conjecture 2; Section 3, Theorem 4 and Conjectures 3 and 4.