Proof of a q-supercongruence conjectured by Guo and Schlosser

TL;DR

This paper proves a q-supercongruence conjecture by Guo and Schlosser involving sums of q-analogues and cyclotomic polynomials, confirming a specific congruence relation for odd integers.

Contribution

The paper provides a rigorous proof of a previously conjectured q-supercongruence involving q-series and cyclotomic polynomials, advancing understanding in q-analogue supercongruences.

Findings

Confirmed the conjectured q-supercongruence for all odd n>1.

Established the congruence relation modulo cyclotomic polynomials.

Extended the theory of q-supercongruences in number theory.

Abstract

In this paper, we confirm the following conjecture of Guo and Schlosser: for any odd integer $n>1$ and $M=(n+1)/2$ or $n-1$, $$ \sum_{k=0}^{M}[4k-1]_{q^2}[4k-1]^2\frac{(q^{-2};q^4)_k^4}{(q^4;q^4)_k^4}q^{4k}\equiv (2q+2q^{-1}-1)[n]_{q^2}^4\pmod{[n]_{q^2}^4\Phi_n(q^2)}, $$ where $[n]=[n]_q=(1-q^n)/(1-q),(a;q)_0=1,(a;q)_k=(1-a)(1-aq)\cdots(1-aq^{k-1})$ for $k\geq 1$ and $\Phi_n(q)$ denotes the $n$-th cyclotomic polynomial.