Some New Explicit Values of Quotients of Theta-Function and Applications to Ramanujan's Continued Fractions

We find some new explicit values of the parameter for positive real numbers and involving Ramanujan's theta-function and give some applications of these new values for the explicit evaluations of Ramanujan's continued fractions. In the process, we also establish two new identities for by using modular equations. 1. Introduction For , Im , define Ramanujan's theta-function as where [1, page 464] is one of the classical theta-functions. In his notebook [2, volume I, page 248], Ramanujan recorded several explicit values of theta-functions and its quotients which are proved by Berndt and Chan [3]. They also found some new explicit values. An account of these can also be found in Berndt's book [4]. Yi [5] also evaluated many new values of by finding explicit values of the parameters and for positive real numbers and which are defined by Yi [5] established several properties of these parameters and found their explicit values by appealing to transformation formulas and theta-function identities for . Recently, Saikia [6] found many new explicit values of quotients of by finding explicit values of the parameter which is a particular case of the parameter where . Saikia [6] also established some new theta-function identities for . In the sequel of the previous work, in this paper we find some new explicit values of the parameters and which are particular cases of the parameter by using some properties of established in [5] and two new theta-function identities for . In addition, we give some applications of these new values of for the explicit evaluations of Ramanujan's continued fractions and defined by The continued fraction is studied by Adiga and Anitha [7]. For explicit evaluations of , see [6]. The continued fraction is called Ramanujan-G llnitz-Gordon continued fraction [4, page 50]. For further references on , see [8, 9]. In Section 2, we record some preliminary results. Section 3 is devoted to prove two new identities for theta-function . In Section 4, we find some new explicit values of the parameter . In Section 5, we evaluate some new values of the parameter . Finally in Section 6, we give applications of these values of and for the explicit evaluations of Ramanujan's continued fractions and . We end the introduction by defining Ramanujan's modular equation. The complete elliptic integral of the first kind is defined by where , denotes the ordinary or Gaussian hypergeometric function, and The number is called the modulus of , and is called the complementary modulus. Let , and denote the complete elliptic integrals of the first kind associated with