Nonelliptic functions from $F(\frac{1}{6},\frac{5}{6};\frac{1}{2};\bullet )$

Abstract

As contributions to the Ramanujan theory of elliptic functions to alternative bases, Li-Chien Shen has developed families of elliptic functions from the hypergeometric functions $F(\frac{1}{3},\frac{2}{3};\frac{1}{2};\bullet )$ and $F(\frac{1}{4},\frac{3}{4};\frac{1}{2};\bullet )$. We apply his methods to the hypergeometric function $F(\frac{1}{6},\frac{5}{6};\frac{1}{2};\bullet )$.