paper

Proof of a q-supercongruence conjectured by Guo and Schlosser

arXiv:2005.14466

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 $n - 1$ , $k = 0 \sum M [4 k - 1]_{q^{2}} [4 k - 1]^{2} \frac{( q ^{- 2} ; q ^{4} ) _{k}^{4}}{( q ^{4} ; q ^{4} ) _{k}^{4}} q^{4 k} \equiv (2 q + 2 q^{- 1} - 1) [n]_{q^{2}}^{4} (mod [n]_{q^{2}}^{4} I^¦_{n} (q^{2})),$ where $[n] = [n]_{q} = (1 - q^{n}) / (1 - q), (a; q)_{0} = 1, (a; q)_{k} = (1 - a) (1 - a q) \dots (1 - a q^{k - 1})$ for $k \geq 1$ and $I^¦_{n} (q)$ denotes the $n$ -th cyclotomic polynomial.