A Four-Genus Kronecker-Limit Evaluation of the Alternating Rogers-Ramanujan Continued Fraction

We evaluate the six odd class-number-four cases left unevaluated in Ramanathan’s treatment of the Rogers-Ramanujan continued fraction. Let $$S(q)=-R(-q),R(q)={\displaystyle \frac{q^{1\mathrm{/}5}}{1+{\displaystyle \frac{q}{1+{\displaystyle \frac{q^2}{1+{\displaystyle \frac{q^3}{1+\cdots }}}}}}}}\mathrm{.}$$ For $n=39,87,111,119,159,287$ we determine the value of $S(e^{-\pi \mathrm{/}\sqrt{5n}})$ by applying the genus-character form of the Kronecker limit formula to the four ideal classes of $\mathbb{Q}(\sqrt{-5n})$. The resulting expressions are given in terms of fundamental units of real quadratic fields. In particular, if $X_n=S(e^{-\pi \mathrm{/}\sqrt{5n}})$, then $$X_n^{-5}+11-X_n^5=\frac{5\sqrt{5}}{U_n},$$ where the six quantities $U_n$ are displayed explicitly below. We also give the corresponding radical expressions and quartic algebraic certificates, together with numerical checks of the six evaluations.