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A further $q$-generalization of the (C.2) and (G.2) supercongruences of Van Hamme

Song-Xiao Li, Su-Dan Wang

Abstract

Applying the $q$-Zeilberger algorithm, we establish a unified $q$-analogue of the (C.2) and (G.2) supercongruences of Van Hamme, which can be viewed as a refinement of several previously known results. As consequences, we obtain a $q$-analogue of supercongruence involving Bernoulli numbers, as well as a refinement of (G.2) supercongruence.

A further $q$-generalization of the (C.2) and (G.2) supercongruences of Van Hamme

Abstract

Applying the -Zeilberger algorithm, we establish a unified -analogue of the (C.2) and (G.2) supercongruences of Van Hamme, which can be viewed as a refinement of several previously known results. As consequences, we obtain a -analogue of supercongruence involving Bernoulli numbers, as well as a refinement of (G.2) supercongruence.

Paper Structure

This paper contains 4 sections, 16 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminary lemmas
  3. Proof of Theorem \ref{['thm:main-1']}
  4. Proof of Theorem \ref{['thm:main-2']}

Key Result

Theorem 1.1

Let $n$ and $d$ be positive integers satisfying $\gcd(n,d)=1$. Let $r$ be an integer with $n+d-nd\leqslant r \leqslant n$ and $n\equiv r \pmod{d}$. Then, modulo $[n]\Phi_n(q)^4$, where $M=(n-r)/d$ or $n-1$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 10 more