Ramanujan recorded four reciprocity formulas for the Rogers–Ramanujan continued fractions. Two reciprocity formulas each are also associated with the Ramanujan–Göllnitz–Gordon continued fractions and a level-13 analog of the Rogers–Ramanujan continued fractions. We show that all eight reciprocity formulas are related to a pair of quadratic equations. The solution to the first equation generalizes the golden ratio and is used to set the value of a coefficient in the second equation; and the solution to the second equation gives a pair of values for a continued fraction. We relate the coefficients of the quadratic equations to important formulas obtained by Ramanujan, examine a pattern in the relation between a continued fraction and its parameters, and use the reciprocity formulas to obtain close approximations for all values of the continued fractions. We highlight patterns in the expressions for certain explicit values of the Rogers–Ramanujan continued fractions by expressing them in terms of the golden ratio. We extend the analysis to reciprocity formulas for Ramanujan’s cubic continued fraction and the Ramanujan–Selberg continued fraction.

中文翻译：

---

Rogers-Ramanujan连续分数的对等公式的性质

Ramanujan记录了Rogers-Ramanujan连续分数的四个互惠公式。两个互惠公式也分别与Ramanujan–Göllnitz–Gordon连续分数和Rogers–Ramanujan连续分数的13级类似物相关。我们表明，所有八个互易公式都与一对二次方程有关。第一个方程的解决方案推广了黄金分割率，并用于设置第二个方程中的系数值。第二个方程的解给出了连续分数的一对值。我们将二次方程的系数与Ramanujan获得的重要公式联系起来，研究了连续分数与其参数之间关系的模式，并使用互易公式获得连续分数所有值的近似值。我们通过用黄金比率表示罗杰斯-拉曼努詹连续分数的某些明确值，在表达式中突出显示模式。我们将分析扩展到Ramanujan立方连续分数和Ramanujan-Selberg连续分数的对等公式。