# Some $q$-supercongruences from transformation formulas for basic hypergeometric series

## TL;DR

This paper develops new $q$-supercongruences using advanced hypergeometric transformation formulas and techniques like microscoping, extending known supercongruences and introducing novel transformation formulas for basic hypergeometric series.

## Contribution

It introduces new $q$-supercongruences derived from transformation formulas and develops a new transformation formula for a nonterminating ${}_{12}phi_{11}$ series.

## Key findings

- Established $q$-analogues of supercongruences by Long and others.
- Derived a new transformation formula for a nonterminating ${}_{12}phi_{11}$ series.
- Generalized Rogers' linearization formula for $q$-ultraspherical polynomials.

## Abstract

Several new $q$-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Wadim Zudilin. More concretely, the results in this paper include $q$-analogues of supercongruences (referring to $p$-adic identities remaining valid for some higher power of $p$) established by Long, by Long and Ramakrishna, and several other $q$-supercongruences. The six basic hypergeometric transformation formulas which are made use of are Watson's transformation, a quadratic transformation of Rahman, a cubic transformation of Gasper and Rahman, a quartic transformation of Gasper and Rahman, a double series transformation of Ismail, Rahman and Suslov, and a new transformation formula for a nonterminating very-well-poised ${}_{12}\phi_{11}$ series. Also, the nonterminating $q$-Dixon summation formula is used. A special case of the new ${}_{12}\phi_{11}$ transformation formula is further utilized to obtain a generalization of Rogers' linearization formula for the continuous $q$-ultraspherical polynomials.