Abstract:Ramanujan recorded four reciprocity formulas for the Roger-Ramanujan continued fractions. Two reciprocity formulas each are also associated with the Ramanujan-Göllnitz-Gordon continued fractions and a level-13 analog of the Roger-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… Show more
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