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Further q-Supercongruences from Singh's Quadratic Transformation

Wei-Wei Qi

TL;DR

This work develops new $q$-supercongruences for truncated ${}_{4}\phi_3$ series by leveraging Singh's quadratic transformation and the creative microscoping method. It establishes parametric $q$-congruences modulo $(1-aq^n)(a-q^n)$, with further corollaries obtained through specialization (e.g., $d=2$ or $d=3$) and $q\to 1$ limits. The proofs synthesize parameter substitutions in Singh's transformation with classical $q$-hypergeometric summations such as $q$-Chu–Vandermonde and Saalschütz, complemented by cyclotomic coprimality arguments. Overall, the results extend existing $q$-analogues of supercongruences and provide a framework for generating additional congruences in the basic hypergeometric setting.

Abstract

In this paper, we investigate some q-congruences for truncated ${}_{4}φ_3$ series by using Singh's quadratic transformation and the creative microscoping method (introduced by Victor J. W. Guo and Zudilin in 2019).

Further q-Supercongruences from Singh's Quadratic Transformation

TL;DR

This work develops new -supercongruences for truncated series by leveraging Singh's quadratic transformation and the creative microscoping method. It establishes parametric -congruences modulo , with further corollaries obtained through specialization (e.g., or ) and limits. The proofs synthesize parameter substitutions in Singh's transformation with classical -hypergeometric summations such as -Chu–Vandermonde and Saalschütz, complemented by cyclotomic coprimality arguments. Overall, the results extend existing -analogues of supercongruences and provide a framework for generating additional congruences in the basic hypergeometric setting.

Abstract

In this paper, we investigate some q-congruences for truncated series by using Singh's quadratic transformation and the creative microscoping method (introduced by Victor J. W. Guo and Zudilin in 2019).
Paper Structure (3 sections, 6 theorems, 39 equations)

This paper contains 3 sections, 6 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. A $q$-Congruence With Parameter $a$
  3. Proofs of the Theorems

Key Result

Theorem 1.1

For a positive $d\geq 2$, a positive integer $n$ with $n \equiv 1 \pmod{2d}$, and an indeterminate $x$, we have Let $d\geq 3$ be an integer, $n$ a positive integer with $n \equiv -1 \pmod{2d}$, and $x$ an indeterminate. Then,

Theorems & Definitions (6)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 3.1