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

## TL;DR

This paper establishes new $q$-congruences for truncated basic hypergeometric series with even powers of $q$, focusing on congruences modulo squares and cubes of cyclotomic polynomials, and explores related conjectures.

## Contribution

It introduces novel $q$-congruences for even power bases and extends previous work on odd powers, including conjectures on higher powers of cyclotomic polynomials.

## Key findings

- New $q$-congruences modulo square and cube of cyclotomic polynomials.
- Complementary results to earlier odd power $q$-congruences.
- Conjectures on higher power $q$-congruences and hypergeometric series.

## Abstract

We provide several new $q$-congruences for truncated basic hypergeometric series with the base being an even power of $q$. Our results mainly concern congruences modulo the square or the cube of a cyclotomic polynomial and complement corresponding ones of an earlier paper containing $q$-congruences for truncated basic hypergeometric series with the base being an odd power of $q$. We also give a number of related conjectures including $q$-congruences modulo the fifth power of a cyclotomic polynomial and a congruence for a truncated ordinary hypergeometric series modulo the seventh power of a prime greater than 3.