An Algorithm to Compute Algebraic Relations between Modular Functions

Supplement material to an article in The Ramanujan Journal by Ralf Hemmecke, Peter Paule, and Silviu Radu.

• Deriving the modular polynomial of $j$ and λ.
• Deriving an identity found by Krishna Alladi (and more) .

• The MultiSamba algorithm has also been used to compute modular polynomials in a fully automatic derivation of several $1/\pi$ formulas, among them the one that was already given by Ramanujan. These formulas where derived in a fully algebraic manner, i.e., no floating point approximation has been used in any intermediate step.

  Demo of the computation (derivation an proof) of \begin{align*} \frac{1}{\pi} &= \frac{2\sqrt{2}}{99^2} \sum_{n=0}^\infty \frac{(4n)!}{n!^4} \frac{26390n+1103}{396^{4n}} \end{align*}

The material presented here is freely available and can even be executed on your computer. To do so install the free computer algebra system FriCAS 1.3.12 (or higher), a Jypyter notebook interface jFriCAS, and QEta 4.0. Look at the installation instructions on the respective packages and contact the authors in case of troubles.