# Dissections of strange $q$-series

## TL;DR

This paper extends divisibility properties of polynomials in the dissections of $q$-series, originally observed in Fishburn numbers, to broader families of $q$-hypergeometric series related to partial theta functions.

## Contribution

It generalizes the divisibility results from specific series to two generic families of $q$-hypergeometric series, confirming their strong divisibility properties.

## Key findings

- Polynomials in dissections are divisible by $q$-factorials.
- Divisibility properties hold for broader $q$-hypergeometric series.
- Results connect series asymptotically to partial theta functions.

## Abstract

In a study of congruences for the Fishburn numbers, Andrews and Sellers observed empirically that certain polynomials appearing in the dissections of the partial sums of the Kontsevich-Zagier series are divisible by a certain $q$-factorial. This was proved by the first two authors. In this paper we extend this strong divisibility property to two generic families of $q$-hypergeometric series which, like the Kontsevich-Zagier series, agree asymptotically with partial theta functions.