A new extension of the (H.2) supercongruence of Van Hamme for primes
$p\equiv 3\pmod{4}$
Abstract
Using Andrews’ multi-series generaliazation of Watson’s
$_8\phi_7$ transformation, we give a new extension of
the (H.2) supercongruence of Van Hamme for primes
$p\equiv 3\pmod{4}$, as well as its
$q$-analogue. Meanwhile, applying the method of ‘creative
microscoping’, recently introduced by the author and Zudilin, we
establish some further $q$-supercongruences modulo
$\Phi_n(q)^3$, where $\Phi_n(q)$
denotes the $n$-th cyclotomic polynomial in $q$.