Properties of reciprocity formulas for the Rogers-Ramanujan continued fractions

TL;DR

This paper explores the properties of reciprocity formulas related to Rogers-Ramanujan and other continued fractions, revealing their connections to quadratic equations, classical constants, and providing new approximations and patterns.

Contribution

It introduces a unified framework linking reciprocity formulas to quadratic equations and extends these results to other Ramanujan continued fractions, offering new insights and approximations.

Findings

All eight reciprocity formulas relate to a pair of quadratic equations.

Solutions to these equations generalize the golden ratio and determine continued fraction values.

Explicit values are expressed in terms of the golden ratio and related constants.

Abstract

Ramanujan recorded four reciprocity formulas for the Roger-Ramanujan continued fraction. Two reciprocity formulas each are also associated with the Ramanujan--G\"ollnitz--Gordon continued fraction and a level-13 analog of the Roger-Ramanujan continued fraction. 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 the pattern of the relation between a continued fraction and its parameters, and use the reciprocity formulas to obtain close approximations for all values of the continued fraction. We highlight patterns in the expressions for certain explicit values of the Rogers-Ramanujan continued fraction by expressing them in terms of the golden ratio. We extend the results to reciprocity formulas for Ramanujan's cubic continued fraction and the Ramanujan-Selberg continued fraction.