Some new formulas for $\pi$

Gert Almkvist, Christian Krattenthaler, Joakim Petersson

We show how to find series expansions for $\pi$ of the form $\pi ={\sum }_{n=0}^{\infty }S(n)/\left(\genfrac{}{}{0ex}{}{mn}{pn}\right)a^n$, where S(n) is some polynomial in $n$ (depending on $m,p,a$). We prov…