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A refined $q$-analogue of some congruences of Van Hamme

Chen Wang, Yu-Chan Tian, Kai Huang

TL;DR

The paper develops a unified q-analogue of Van Hamme's B.2, E.2, and F.2 supercongruences, extending prior q-analogue work and refining several known results. It introduces a q-WZ framework and cyclotomic-modulus techniques to prove the parametric congruence with Euler-polynomial refinements. The main result demonstrates a parametric supercongruence that specializes to classical cases when q tends to 1, and yields corollaries for prime powers. The work enhances understanding of Ramanujan-type congruences in the q realm and provides tools for further parametric refinements via q-WZ and Euler polynomials.

Abstract

In 1997, Van Hamme proposed 13 supercongruences corresponding to $1/π$ series of the Ramanujan-type. Inspired by the recent work of V.J.W. Guo, we establish a unified $q$-analogue of Van Hamme's (B.2), (E.2) and (F.2) supercongruences, which is also a refinement of some known results obtained by different authors. As a consequence, we give a parametric supercongruence related to the Euler polynomials.

A refined $q$-analogue of some congruences of Van Hamme

TL;DR

The paper develops a unified q-analogue of Van Hamme's B.2, E.2, and F.2 supercongruences, extending prior q-analogue work and refining several known results. It introduces a q-WZ framework and cyclotomic-modulus techniques to prove the parametric congruence with Euler-polynomial refinements. The main result demonstrates a parametric supercongruence that specializes to classical cases when q tends to 1, and yields corollaries for prime powers. The work enhances understanding of Ramanujan-type congruences in the q realm and provides tools for further parametric refinements via q-WZ and Euler polynomials.

Abstract

In 1997, Van Hamme proposed 13 supercongruences corresponding to series of the Ramanujan-type. Inspired by the recent work of V.J.W. Guo, we establish a unified -analogue of Van Hamme's (B.2), (E.2) and (F.2) supercongruences, which is also a refinement of some known results obtained by different authors. As a consequence, we give a parametric supercongruence related to the Euler polynomials.
Paper Structure (2 sections, 12 theorems, 62 equations)

This paper contains 2 sections, 12 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['mainth']}

Key Result

Theorem 1.1

Let $n$ and $d$ be positive integers with $\gcd(n,d)=1$. Let $r$ be an integer with $\gcd(r,d)=1$ and $m$ be the least nonnegative residue of $-r/d$ modulo $n$. Then where $M=m$ or $M=n-1$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Corollary 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • ...and 11 more