Level $17$ Ramanujan-Sato series

TL;DR

This paper introduces new Ramanujan-Sato series for 1/π using level 17 modular functions, expanding the class of such series with modular identities similar to those at levels 5 and 13.

Contribution

It constructs a novel class of Ramanujan-Sato series for 1/π based on level 17 modular functions and identities, extending previous work at levels 5 and 13.

Findings

Derived new Ramanujan-Sato series for 1/π

Identified modular identities at level 17

Provided list of rational and quadratic series for singular values

Abstract

Two level 17 modular functions $$ r=q^{2}\prod_{n=1}^{\infty}(1-q^{n})^{\left(\frac{n}{17}\right)},\quad s=q^{2}\prod_{n=1}^{\infty}\frac{(1-q^{17n})^{3}}{(1-q^{n})^{3}}, $$ are used to construct a new class of Ramanujan-Sato series for $1/\pi$. The expansions are induced by modular identities similar to those level of 5 and 13 appearing in Ramanujan's Notebooks. A complete list of rational and quadratic series corresponding to singular values of the parameters is derived.