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

TL;DR

This paper establishes new theorems connecting Ramanujan-type series for 1/π and π^2 with modular functions and Calabi-Yau differential equations, providing a matrix-based framework for these series.

Contribution

It introduces a matrix formulation of Ramanujan-type series for 1/π and π^2, linking them to modular functions and Calabi-Yau equations, advancing theoretical understanding.

Findings

Proves new theorems for $_3F_2$ Ramanujan-type series for 1/π.

Develops connections between $_5F_4$ series for 1/π^2 and Calabi-Yau differential equations.

Provides a matrix framework for analyzing these series.

Abstract

In this paper we prove theorems related to the Ramanujan-type series for $1/\pi$ (type $_3F_2$) and to the Ramanujan-like series, discovered by the author, for $1/\pi^2$ (type $_5F_4$). 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.