Bilateral Ramanujan-like series for 1/πk and their congruences

Abstract

We prove a kind of bilateral series that terminates to one side (left or right) related to Ramanujan-like series for negative powers of [Formula: see text], and conjecture a type of supercongruences associated with them. We support this conjecture by checking all the cases for many primes. In addition, we are able to prove a few of them from some terminating hypergeometric identities. Finally, we make an intriguing observation.