Some New Explicit Values of Quotients of Ramanujan’s Theta Functions and Continued Fractions

We evaluate some new explicit values of quotients of Ramanujan’s theta functions and use them to find explicit values of Ramanujan’s continued fractions. 1. Introduction Ramanujan’s general theta function is defined by Two important special cases of are the theta functions and [1, page 36, Entry 22] defined by, for , where In his notebooks [2], Ramanujan recorded many explicit values of theta functions and . All these values were proved by Berndt [3, page 325] and Berndt and Chan [4]. Yi [5] introduced the parameter for positive real numbers and defined by and used the particular case to find explicit values . Baruah and Saikia [6] defined the parameter for positive real numbers and as and used the particular case to find explicit values . Saikia [7] also established some explicit values . In this paper, we consider the particular cases and of the parameters and , respectively. By using theta function identities, we find some new explicit values of the parameters and . Particularly, we evaluate and for 3/2, 2/3, 6, 1/6, 5/2, 2/5, 10, and 1/10. Previously, Yi [5] evaluated for 1, 3, 1/3, 9, 1/9, 5, 1/5, 25, and 1/25. Saikia [8] evaluated for 2, 1/2, 4, 1/4, 7, 1/7, 49, and 1/49. Baruah and Saikia [6] evaluated for 1, 3, 1/3, 9, 1/9, 5, 1/5, 25, 1/25, 7, 1/7, 13, 1/13, 49, and 1/49. As an application to our new values, we evaluate some old and new explicit values of Ramanujan’s cubic continued fraction and a continued fraction of order twelve which are, respectively, defined by, for , The continued fraction was recorded by Ramanujan on page 366 of lost notebook [9]. We refer to [10–14] for explicit evaluations of . The continued fraction was introduced by Naika et al. [15]. We refer to [8, 15] for explicit evaluations of . The presentation of the paper is as follows. In Section 2 we record some preliminary results for ready references in this paper. Section 3 is devoted to explicit evaluations of the parameters and . In Section 4 we use new explicit values of and to evaluate some explicit values of the continued fractions and . We end this introduction by noting the following remarks regarding and from [6, page 1764, Theorem 4.1] and [5, page 385, Remark 2.3], respectively. Remark 1. The parameter has positive real value with and increases as increases. Remark 2. The parameter has positive real value with and the values of decrease as increases. 2. Preliminary Results Lemma 3 (see [16, Theorem 3.2]). If and , then Lemma 4 (see [16, Theorem 3.1]). If and , then Lemma 5 (see [16, Theorem 3.2]). If and , then Lemma 6 (see [16, Theorem 3.9]). If and , then