Series for $1/\pi$ of signature 20

TL;DR

This paper uses classical theta and Eisenstein series to derive differential equations and expansions for modular forms of level 20, leading to new formulas for 1/π of signature 20 inspired by Ramanujan.

Contribution

It introduces new differential equations and series expansions for modular forms of level 20, extending Ramanujan's work on 1/π formulas.

Findings

Derived differential equations for level 20 modular forms

Established new series expansions for 1/π of signature 20

Connected classical theta functions to modern modular form theory

Abstract

Properties of theta functions and Eisenstein series dating to Jacobi and Ramanujan are used to deduce differential equations associated with McKay Thompson series of level 20. These equations induce expansions for modular forms of level 20 in terms of modular functions.The theory of singular values is applied to derive expansions for $1/\pi$ of signature $20$ analogous to those formulated by Ramanujan.