A -SUPERCONGRUENCE MODULO THE THIRD POWER OF A CYCLOTOMIC POLYNOMIAL

Abstract

AbstractWe derive a q-supercongruence modulo the third power of a cyclotomic polynomial with the help of Guo and Zudilin’s method of creative microscoping [‘A q-microscope for supercongruences’, Adv. Math.346 (2019), 329–358] and the q-Dixon formula. As consequences, we give several supercongruences including $$ \begin{align*}\sum_{k=0}^{(p-2)/3}\frac{(\frac{2}{3})_k^3}{(1)_k^3}\equiv\frac{p}{2}\frac{(1)_{(p-2)/3}}{(\frac{4}{3})_{(p-2)/3}}\pmod{p^3},\end{align*} $$ where p is a prime with $p\equiv 5\pmod {6}$ .