New theta function identities for a continued fraction of Ramanujan and their applications

Abstract

We prove three new theta function identities for the continued fraction H(q) defined by $$\begin{aligned} H(q):=q^{1/8}-\frac{q^{7/8}}{1-q}_{+}\frac{q^2}{1+q^2}_{-}\frac{q^3}{1-q^3}_{+}\frac{q^4}{1+q^4}_{- \cdots }, \vert q\vert <1. \end{aligned}$$ The theta-function identities are then used to prove integral representations for the continued fraction H(q). We also prove general theorems and reciprocity formulas for the explicit evaluation of the continued fraction H(q). The results are analogous to those of the famous Rogers-Ramanujan continued fraction.