Andrews–El Bachraoui positivity conjecture for the series F_k,1(q)

Let $k$ be a positive integer and define

F_{k,1}(q):=\sum_{n\ge 0}\frac{(q^{2n+2},q^{2n+2k};q^2)_\infty}{(q^{2n+1};q^2)_\infty^2}q^{2n},

where $(A;q)_n:=\prod_{j=0}^{n-1}(1-Aq^j)$ and $(A_1,A_2;q)_n:=(A_1;q)_n(A_2;q)_n$ . A power series is positive if every coefficient is strictly positive.

Andrews–El Bachraoui positivity conjecture. For every positive integer $k$ , the series $F_{k,1}(q)$ is positive.

The series arises as the generating function for the difference between the enumerations of two disjoint subsets of a special family of bicolored partitions; positivity would therefore imply an inequality between those two partition enumerations. The paper proves this conjecture.

The conjecture asserts that the generating series is positive for every positive integer parameter. The retrieved literature records earlier partial results and now includes a paper claiming a complete proof.

Known results

Andrews and Bachraoui established strict positivity for $1 \le k \le 4$ .
The paper published January 7, 2026 proves nonnegativity for $5 \le k \le 7$ analytically.
The same paper verifies nonnegativity computationally for $k \in \{8,9,10\}$ .
A broader July 12, 2025 study records positivity for some initial parameter values without resolving the full family.

Claimed complete proof (date not stated)

The paper Proof of the Andrews–El Bachraoui positivity conjecture claims positivity for every positive integer $k$ , using $q$ -hypergeometric methods, a reformulation of $F_{k,1}(q)$ , and an auxiliary finite $q$ -series positivity theorem. The claim is unverified in the retrieved sources.

Current status (as of September 2026): Earlier cases and finite computations are established, while a complete proof is claimed in an arXiv paper but remains unverified.

Andrews–El Bachraoui positivity conjecture for the series $F_{k,1}(q)$

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Progress summary

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