Dissections of a "strange" function

TL;DR

This paper investigates the properties of Kontsevich and Zagier's 'strange' function, proving a generalization of a conjecture about its polynomial dissections and reestablishing related congruences for Fishburn numbers.

Contribution

It proves a generalized form of a conjecture on polynomial dissections of the 'strange' function and rederives known Fishburn number congruences using this framework.

Findings

Proved a generalized conjecture on polynomial dissections of the 'strange' function.

Reproved known congruences for Fishburn numbers modulo prime powers.

Abstract

The "strange" function of Kontsevich and Zagier is defined by \[F(q):=\sum_{n=0}^\infty(1-q)(1-q^2)\dots(1-q^n).\] 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 $\xi(n)$ (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 $\xi(n)$ modulo prime powers.