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

We prove general theorems for the explicit evaluations of the level 13 analogue of Rogers-Ramanujan continued fraction and find some new explicit values. This work is a sequel to some recent works of S. Cooper and D. Ye. 1. Introduction For and , the Dedekind eta function and Ramanujan’s function are defined by where . The famous Rogers-Ramanujan continued fraction is defined by 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 . These values are first proved by Watson [5, 6] and Ramanathan [7]. For further references on explicit evaluations of see [8–14]. The Rogers-Ramanujan continued fraction is closely related to the function by the following two beautiful relations: which are stated by Ramanujan [15, page 267, (11.6)] and first proved by Watson [5]. In his second notebook [2, 15] Ramanujan stated an interesting analogue of . If then Proofs of (5) can be found in [16, 17]. The function is studied by Cooper and Ye [18, 19]. In [18] Cooper and Ye evaluated or indicated the explicit values of and for = 1, 2, 3, 5, 7, 9, 13, 15, 31, 55, 69, 129, 231, and 255 by using the methods of reciprocity formulas, modular equations, Ramanujan-Weber class invariant, and Kronecker’s limit formula and in terms of the functions and [18, page 94, (1.7) and (1.8)] which are defined by From (6) and (7), it is clear that if we know or for any particular value of , then can be determined by solving the corresponding quadratic equations. Cooper and Ye [18, page 104, Theorem 4.1] further established that where and . They also indicated that if we know for any particular , then and can be determined by appealing to (8) and therefrom and can be determined at the same . For this they evaluated for = 5, 9, 13, 69, and 129. 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 . 2. Preliminaries Lemma 1 (see [15, page 43, Entry 27(iii), (iv), (vi)]). If and are such that the modulus of each exponential argument below is less than 1 and , then Lemma 2