Nonnegative coefficients for polynomial fake Gaussian sequences

Prove that for given positive integers m and n and every symmetric sequence of non-negative integers \((a_1,a_2,\ldots,a_n)\), if the expression \(\prod_{i=1}^{n}\frac{(1-q^{m+i})^{a_i}}{(1-q^i)^{a_i}}\) is a polynomial in q, then it has non-negative coefficients.

Background

This conjecture extends the q-binomial positivity conjecture to D. Stanton’s fake Gaussian sequences. The sequence of exponents is required to be symmetric and consist of non-negative integers, while the polynomiality condition is imposed only for the specified value of m.

The authors explain that Stanton’s original conjecture requires the expression to be a polynomial for every positive integer m, whereas Conjecture 2 drops that stronger polynomiality requirement. They report computational evidence for randomly generated symmetric sequences and verify several structured special cases, but do not prove the general assertion.

References

Conjecture 2. Let m and n be given positive integers, and let (a1, a2, . . . , an) be a symmetric sequence of non-negative integers. If the expression (2) is a polynomial (in q) then it has non-negative coefficients.

— A positivity conjecture for a quotient of $q$-binomial coefficients  (2502.06032 - Gatzweiler et al., 9 Feb 2025) in Conjecture 2, Remark (2) following Conjecture 1, Section 1, p. 2

Moreover, a direct translation of Conj. 1.3 states that

\prod_{i=1}n \left({[m+i]}/{[i]}\right){a_i} \in \Phi+

when the expression considered is polynomial.

— A $(q,t)$-Overview of $q$-Analogs  (2608.30979 - Bergeron, 31 Aug 2026) in Section 7, subsection “Cyclotomic generating functions,” equations (question_fract_m_ai) and surrounding paragraph