2014/08/06 by Scott Ahlgren, Ahlgren, Scott, Byungchan Kim +1
Mathematics · #05A19 #11F20 #11P83 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05A19 #msc:11F20 #msc:11P83
paper · pdf · doi:10.48550/arxiv.1408.1334
arxiv created 2014/08/06 · openalex publication_date 2014/08/06 · arxiv updated 2014/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The "strange" function of Kontsevich and Zagier is defined by F(q):=∑n=0^∞(1-q)(1-q2)…(1-qn). This series is defined only when q is a root of unity, and provides an example of what Zagier has called a "quantum modular form." In their recent work on congruences for the Fishburn numbers ξ(n) (whose generating function is F(1-q)), Andrews and Sellers recorded a speculation about the polynomials which appear in the dissections of the partial sums of F(q). We prove that a more general form of their speculation is true. The congruences of Andrews-Sellers were generalized by Garvan in the case of prime modulus, and by Straub in the case of prime power modulus. As a corollary of our theorem, we reprove the known congruences for ξ(n) modulo prime powers.