Parametric Evaluations of the Rogers-Ramanujan Continued Fraction

In this paper with the help of the inverse function of the singular moduli we evaluate the Rogers-Ranmanujan continued fraction and its first derivative. 1. Introductory Definitions and Formulas For , the Rogers-Ramanujan continued fraction (RRCF) (see [1]) is defined as We also define Ramanujan give the following relations which are very useful: From the theory of elliptic functions (see [1–3]), is the elliptic integral of the first kind. It is known that the inverse elliptic nome , is the solution of the equation where . When is rational then the are algebraic numbers. We can also write the function using elliptic functions. It holds (see [3]) and also holds From [4] it is known that Consider now for every the equation which has solution Hence for example With the help of function we evaluate the Rogers Ramanujan continued fraction. 2. Propositions The relation between and is (see [1] page 280) For to solve (2.1) we give the following. Proposition 2.1. The solution of the equation when one knows is given by where If it happens that and , then and , . Proof. The relation (2.3) can be found using Mathematica. See also [5]. Proposition 2.2. If and then where is root of . Proof. Suppose that , where is positive integer and is positive real then it holds that where The following formula for is known: Thus, if we use (1.4) and (1.7) and the above consequence of the theory of elliptic functions, we get: See also [4, 5]. 3. The Main Theorem From Proposition 2.2 and relation we get Combining (2.2) and (3.1), we get Solving with respect to , we get Also we have The above equalities follow from [1] page 280 Entry 13-xii and the definition of . Note that is the multiplier. Hence for given , we find and we get the following parametric evaluation for the Rogers Ramanujan continued fraction Thus for a given we find and from (2.4) and (2.5). Setting the values of , , in (2.3) we get the values of and (see Proposition 2.1). Hence from (3.5) if we find we know . The clearer result is as follows. Main Theorem 3. When is a given real number, one can find from (2.3). Then for the Rogers-Ramanujan continued fraction the following holds: Theorem 3.1. (the first derivative). One has Proof. Combining (1.7) and (1.9) and Proposition 2.2 we get the proof. We will see now how the function plays the same role in other continued fractions. Here we consider also the Ramanujan's Cubic fraction (see [5]), which is completely solvable using . Define the function Set for a given Then as in Main Theorem, for the Cubic continued fraction , the following holds (see [5]): Observe here that