Proof of a supercongruence conjectured by Sun through a $q$-microscope

Abstract

Recently, Z.-W. Sun made the following conjecture: for any odd prime $p$ and odd integer $m$, $$\frac{1}{m^2\left(\genfrac{}{}{0ex}{}{m-1}{(m-1)/2}\right)}(\sum _{k=0}^{(pm-1)/2}\frac{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}{8^k}-\left(\frac{2}{p}\right)\sum _{k=0}^{(m-1)/2}\frac{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}{8^k})\equiv 0(\mathrm{mod}{p}^2).$$ In this note, applying the "creative microscoping" method, introduced by the author and Zudilin, we confirm the above conjecture of Sun.

6 pages

References in corpus (1)

• Some new $q$-congruences for truncated basic hypergeometric series: even powers