New General Theorems and Explicit Values of the Level 13 Analogue of Rogers-Ramanujan Continued Fraction

This continued fraction was introduced by Rogers [1] in 1894 and rediscovered by Ramanujan in approximately 1912. In his notebooks [2], lost notebook [3], and his first two letters to Hardy [4], Ramanujan recorded several explicit values of R(q). These values are first proved by Watson [5, 6] and Ramanathan [7]. For further references on explicit evaluations ofR(q) see [8–14]. The Rogers-Ramanujan continued fractionR(q) is closely related to the functionf(−q) by the following two beautiful relations:

In this paper, we prove some general theorems for the explicit evaluations () and (−) by parameterizations of Dedekind eta function and find some old and new explicit values.In Section 2, we record some preliminary results which will be used in the subsequent sections.In Section 3, we prove general theorems for the explicit evaluations of (±) and evaluate some old and new explicit values.In Section 4, we find some new explicit values of () by parametrization.Finally, in Section 5, we consider the function ().

General Theorems for Explicit Evaluations of 𝑅(±𝑞)
In this section we prove general theorems for the explicit evaluations of (±) and find some explicit values.
Theorem 4. One has the following.
Theorem 5.If   and   are as defined in Theorem 4, then Proof.We use the definitions of   and   and use (10) and (11), respectively, to complete the proof.

Corollary 6.
If   and   are as defined in Theorem 4, then Proof.We replace  by 1/ in Theorem 4(i) and (ii) and employ Theorem 5 to arrive at (i) and (ii), respectively.
From Theorem 4(i) and Corollary 6(i), it is clear that if we know the explicit values of the parameter   , then explicit values of ( −2√/13 ) and ( −2/√13 ) can be evaluated, respectively.Similarly, if we know the explicit values of the parameter   , then explicit values of (− −√/13 ) and (− −/√13 ) can be determined from Theorem 4(ii) and Corollary 6(ii), respectively.
Next we find some explicit values of the parameters   and   .

Corollary 7. One has
Proof.We set  = 1 in Theorem 5 to complete the proof.
Theorem 9.One has Proof.We use the definition of   in Lemma 2 to complete the proof.
Solving (28) and choosing the appropriate root, we complete the proof.
Theorem 12.One has Proof.We employ the definition of   in Lemma 3 to complete the proof.(32) Proof.Setting  = 1 in Theorem 12 and using the result  1 = 1, we obtain Solving (33) and choosing the appropriate root, we arrive at (i).Again, setting  = 1/3 in Theorem 12 and simplifying using Theorem 5, we obtain 2 √ 13 =  4  3 + 3 Solving (34) and choosing the appropriate root, we complete the proof of (ii).
To prove (ii) we set  = 1/3, employ the result  1/ = 1/  from Theorem 5, and solve the resulting equation.We complete the proof.

Explicit Evaluations of 𝑉(𝑞)
In this section we evaluate explicit values of the function () defined in (9) by parametrization method.Cooper and Ye [18] used Ramanujan-Weber class invariants to evaluate ().

Explicit Evaluations of 𝐹(𝑞)
Theorem 24.If   is as defined in Theorem 4, then