Bilateral Ramanujan-like series for $1/\pi^k$ and their congruences

TL;DR

This paper introduces bilateral Ramanujan-like series for powers of 1/π, explores related supercongruences, and provides partial proofs and extensive numerical evidence for these conjectures.

Contribution

It establishes new bilateral series related to Ramanujan-like series and proposes conjectures on supercongruences, supported by computational verification and some proven cases.

Findings

Proved certain supercongruences for primes

Discovered bilateral series related to 1/π^k

Numerical evidence supporting conjectures

Abstract

We prove a kind of bilateral semi-terminating series related to Ramanujan-like series for negative powers of $\pi$, and conjecture a type of supercongruences associated to 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.