2021/08/09 by Liuquan Wang, Wang, Liuquan · 1 citation
Mathematics · Physics and Astronomy · #11F03 #11F30 #11P55 #26D15 #26D20 #Advanced Differential Equations and Dynamical Systems #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2108.03932
openalex publication_date 2021/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We denote by ct(m)(n) the coefficient of qn in the series expansion of (q;q)_∞m(qt;qt)_∞-m, which is the m-th power of the infinite Borwein product. Let t and m be positive integers with m(t-1)≤ 24. We provide asymptotic formula for ct(m)(n), and give characterizations of n for which ct(m)(n) is positive, negative or zero. We show that ct(m)(n) is ultimately periodic in sign and conjecture that this is still true for other positive integer values of t and m. Furthermore, we confirm this conjecture in the cases (t,m)=(2,m),(p,1),(p,3) for arbitrary positive integer m and prime p.