# Some new $q$-congruences for truncated basic hypergeometric series

## TL;DR

This paper establishes new $q$-congruences for truncated basic hypergeometric series, including results modulo squares and cubes of cyclotomic polynomials, using advanced combinatorial and algebraic techniques.

## Contribution

It introduces novel $q$-congruences for hypergeometric series, expanding the understanding of their modular properties and providing parametric generalizations.

## Key findings

- Congruences modulo the square of cyclotomic polynomials
- Congruences modulo the cube of cyclotomic polynomials
- Conjectures on higher power congruences

## Abstract

We provide several new $q$-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric generalizations thereof. These are established by a variety of techniques including polynomial argument, creative microscoping (a method recently introduced by the first author in collaboration with Zudilin), Andrews' multiseries generalization of the Watson transformation, and induction. We also give a number of related conjectures including congruences modulo the fourth power of a cyclotomic polynomial.