Some new formulas for $\pi$

Abstract

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 prove that there exist such expansions for $m=8k$, $p=4k$, $a=(-4{)}^k$, for any $k$, and give explicit examples for such expansions for small values of $m$, $p$ and $a$.

28 pages, LaTeX; some important references added