New type series for 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.