Yazan Alamoudi

Affiliation: University of Florida

Title Of Talk: A resolution of a conjecture related to the Alladi-Schur polynomials

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Abstract: In 2016, in a preprint which was ultimately collected for ALLADI60 and published in 2018, G. Andrews conjectured that the polynomial \[c(n,j)=\sum_{r=0}^j\sum_{0\leq 3i\leq r}\frac{(-1)^iq^{4nj-2nr+j+3i(i-1)}(q^2;q^2)_n}{(q^2;q^2)_{n-j}(q^2;q^2)_{j-r}(q^2;q^2)_{r-3i}(q^6;q^6)_i},\] has only nonnegative coefficients. Andrews was investigating such polynomials because of their relation to the Alladi-Schur polynomials, which played a key role in his refinement of the Alladi-Schur theorem. Near the end of last year, I resolved this conjecture using a completely elementary technique. In this talk, I will detail the proof and, if time permits, discuss further lower bounds, as well as highlight intriguing intricacies and potential pitfalls associated with this question. Additionally, I will provide remarks on the connection to Andrews' refinement of the Alladi-Schur theorem."

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