Some New Explicit Values of Parameters and of Quotients of Eta-Function

We find some new explicit values of the parameters and of quotients of eta-function by using Ramanujan's class invariants. 1. Introduction For and , the Dedekind eta-function and Ramanujan’s function are defined by where . In page 212 of his lost notebook [1], Ramanujan defined and provided a list of eleven recorded and ten unrecorded values of for positive integers . All 21 values of and many more were established by Berndt et al. [2] by using the modular -invariant, modular equations, Kronecker’s limit formula, and the explicit Shimura reciprocity law. An account of this can also be found in Chapter 9 of [3]. Closely related to is the parameter introduced by Ramanathan [4] and is defined as Yi [5, 6] also found several values of parameters and by finding the explicit values of her parameters and , defined by where and are positive real numbers. In fact, , and . Baruah and Saikia [7, 8] also evaluated several new values of and . A generalization of has been studied by Naika et al. [9]. Here we give a list of the values of for which and were evaluated in [1, 2, 5–8] without referring them specifically to avoid repetitions. The values of for which were evaluated in the literature are = 1, 3, 5, 7, 9, 11, 13, 15, 5/3, 17, 25, 33, 41, 49, 57, 65, 81, 89, 73, 97, 121, 169, 193, 217, 241, 265, 289, 361, 11/3, and 19/3, and the values of values of for which were evaluated in literature are = 1, 2, 3, 4, 5, 6, 3/2, 7, 8, 9, 10, 5/2, 11, 13, 14, 7/2, 15, 5/3, 17, 19, 20, 5/4, 22, 11/2, 25, 26, 13/2, 44, 11/4, and 49. We also note from [2, page 281] and [7, page 43, Theorem 4.2] that respectively. In this paper we find further new values of and by using Ramanujan’s class invariants which are defined as where . In particular, we evaluate values of for = 23, 31, 47, 59, 71, 2, 4, 6, 10, 5/2, 14, and 7/2 and new values of for = 23, 31, 47, 59, and 71. It worth to mention here that for the first time in this paper explicit values of for some even values of are evaluated. Previously, Berndt et al. [2] calculated explicit values of , , and by using Ramanujan’s class invariants. An account of Ramanujan’s class invariants can be found in Berndt’s book [10]. For further references on Ramanujan’s class invariants refer see [11–14]. The explicit values of the functions and can be applied to find explicit values of Ramanujan’s cubic continued fraction defined by We refer to [7] for details. To this end we define a general parameter [14, page 2, equation (10)] for all positive real numbers and and connected with parameters , , and Ramanujan’s class invariants as Saikia [14]