On some continued fraction expansions for the ratios of the function

Abstract

In his lost notebook, Ramanujan has defined the function $\rho(a,~b)$ by \begin{equation*} \rho(a ,~b) := \left(1 +\frac{1}{b}\right)\sum_{n=0}^{\infty} \frac{(-1)^{n}q^{n(n+1)/2} {a}^{n}{b}^{-n}}{(-aq)_{n}}, \end{equation*} where $|q|< 1,$ and $(a; q)_n = \prod_{k = 0}^{n-1}(1-aq^k),~~~n=1,2,3,\dots,$ and has given a beautiful reciprocity theorem involving $\rho(a,~b)$. In this paper we obtain some continued fraction expansions for the ratios of $\rho(a,~b)$ with some of its contiguous functions. We also obtain some interesting special cases of our continued fraction expansions which are analogous to the continued fraction identities stated by Ramanujan. \end