# Factors of some truncated basic hypergeometric series

## TL;DR

This paper proves that specific truncated basic hypergeometric series contain a cyclotomic polynomial factor, confirming recent conjectures and proposing new conjectures on related q-congruences.

## Contribution

It establishes the presence of cyclotomic polynomial factors in truncated hypergeometric series and confirms two recent conjectures, advancing understanding of q-series properties.

## Key findings

- Truncated basic hypergeometric series have the factor _n(q)^2.
- Confirmed two conjectures regarding these series and cyclotomic polynomials.
- Proposed new conjectures on q-congruences modulo _n(q)^2.

## Abstract

We prove that certain basic hypergeometric series truncated at $k=n-1$ have the factor $\Phi_n(q)^2$, where $\Phi_n(q)$ is the $n$-th cyclotomic polynomial. This confirms two recent conjectures of the author and Zudilin. We also put forward some conjectures on $q$-congruences modulo $\Phi_n(q)^2$.