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 $$\sum _{k=0}^{\infty }(198k^2-425k+210)\frac{k^3{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^3}{4096^k}=-\frac{1}{21\pi }$$ and $$\sum _{k=0}^{\infty }\frac{198k^2-227k+47}{{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^3}=\frac{3264-4{\pi }^2}{63}.$$ We also pose many conjectures in this new direction.

51 pages, accepted version for publication

References in corpus (2)

• Open Conjectures on Congruences
• Ramanujan series upside-down