On Ramanujan's cubic continued fraction as a modular function
The authors study Ramanujan's cubic continued fraction as a modular function. For a complex number \(\tau \in \mathbb{U}\), the Rogers-Ramanujan continued fraction is defined by \[ r(\tau )=\frac{q^{1/5}}{1+\frac{q}{1+\frac{q^{2}}{1+\frac{q^{3}}{1+\cdots}}}} =q^{1/5}\prod\limits_{n=1}^{\infty }(1-q^{n})^{(\frac{n}{5})}, \] where \(q=e^{2\pi i\tau }\) and \((\frac{n}{5})\) is the Legendre function. Ramanujan proved that \[ r(i)=\sqrt{\frac{5+\sqrt{5}}{2}}-\frac{\sqrt{5}+1}{2}\;\;\;\text{and} \;\;r(\frac{5+i}{2})=\sqrt{\frac{5-\sqrt{5}}{2}}-\frac{\sqrt{5}-1}{2}. \] Later in 2005 W. Duke mentioned that Ramanujan's cubic continued fraction \(C(\tau )\) defined as \[ C(\tau )=\frac{q^{1/3}}{1+\frac{q+q^{2}}{1+\frac{q^{2}+q^{4}}{1+\frac{q^{3}+q^{6}}{1+\cdots}}}}=q^{1/3}\prod\limits_{n=1}^{\infty }\frac{(1-q^{6n-1})(1-q^{6n-5})}{(1-q^{6n-3})^{3}} \] with modularity for \(\Gamma (6)\). They prove some nice results on Ramanujan's cubic continued fraction.
scientific article; zbMATH DE number 5856432
scientific article