New type series for powers of $\pi$

TL;DR

This paper introduces two novel series involving powers of pi inspired by Ramanujan and Zeilberger, providing proofs for specific series and proposing numerous conjectures in this new area.

Contribution

It presents new types of series for powers of pi, proves two specific series, and explores a novel research direction with many conjectures.

Findings

Proved a series for 1/pi involving binomial coefficients.

Proved a series for a combination of pi^2 and rational numbers.

Established a new framework for series related to powers of pi.

Abstract

Motivated by Ramanujan-type series and Zeilberger-type series, in this paper we investigate two new types of series for powers of $\pi$. For example, we prove that $$\sum_{k=0}^\infty(198k^2-425k+210)\frac{k^3\binom{2k}k^3}{4096^k}=-\frac1{21\pi}$$ and $$\sum_{k=0}^\infty\frac{198k^2-227k+47}{\binom{2k}k^3}=\frac{3264-4\pi^2}{63}.$$ We also pose many conjectures in this new direction.