A Matrix form of Ramanujan-type series for $1/\pi$

Abstract

In this paper we prove theorems related to the Ramanujan-type series for $1/\pi$ (type ${}_3{F}_2$) and to the Ramanujan-like series, discovered by the author, for $1/{\pi }^2$ (type ${}_5{F}_4$). Our developments for the cases ${}_3{F}_2$ and ${}_5{F}_4$ connect with the theory of modular functions and with the theory of Calabi-Yau differential equations, respectively.