Modular Relations and Explicit Values of Ramanujan-selberg Continued Fractions

By employing a method of parameterizations for Ramanujan's theta-functions, we find several modular relations and explicit values of the Ramanujan-Selberg continued fractions .

In his lost notebook, Ramanujan [5, page 44] also stated that if |q| < 1 and (1.12) From (1.5) and (1.9)-(1.12),we easily see that (1.13)By setting we also note that (1.15) N. D. Baruah and N. Saikia 3 In this paper, we find several modular relations connecting the above continued fractions in different arguments.We present these in Sections 3-5.
We observe that Vasuki and Shivashankara [7] had found explicit values of H(e −π √ n ) for n = 3,1/3,5,1/5,7,1/7,13, and 1/13 by using eta-function identities and transformation formulas.In Sections 6 and 7, we also find several new explicit values of H(e −π √ n ) by using the parameter J n , defined by where n is any positive real number.We note that the parameter J n is equivalent to the parameter r 4,n , which is a particular case of the parameter r k,n , introduced by Yi [10, page 4, equation (1.11)] (see also [9, page 11, equation (2.1.1)]),and defined by where n and k are positive real numbers.We note that Zhang [11, page 11, Theorems 2.1 and 2.2] established general formulas for explicit evaluations of S 1 (e −π √ n ) and T(e −π √ n ) in terms of Ramanujan's singular moduli.In fact, he proved that where q = e −π √ n and the singular modulus α n is that unique positive number between 0 and 1 satisfying In Section 8, we establish general formulas for explicit evaluations of S 1 (e −π √ n ) and S 1 (e −π/ √ n ) in terms of the parameter r k,n .We also give some particular examples.Since Ramanujan's modular equations are central in our evaluations, we now give the definition of a modular equation as given by Ramanujan.Let K, K := K(k ), L, and L := L(l ) denote the complete elliptic integral of the first kind associated with the moduli k, k := √ 1 − k 2 , l, and l := √ 1 − l 2 , respectively.Suppose that the equality holds for some positive integer n.Then a modular equation of degree n is a relation between the moduli k and l which is implied by (1.21).
If we set 4 Ramanujan-Selberg continued fractions we see that (1.21) is equivalent to the relation q n = q .Thus, a modular equation can be viewed as an identity involving theta-functions at the arguments q and q n .Ramanujan recorded his modular equations in terms of α and β, where α = k 2 and β = l 2 .We say that β has degree n over α.The multiplier m connecting α and β is defined by (1.23) If q = exp(−πK /K), one of the most fundamental relations in the theory of elliptic functions is given by the formula [1, pages 101-102] So the multiplier m of degree n defined in (1.23) can also be written as where z r = φ 2 (q r ).Again, if we put q = exp(−πK /K), z = z 1 , and x = α in [1, Entries 10(i), 11(i), 12(i), 12(ii), and 12(iv), pages 122-124], then we have the representations , (1.27) , (1.29) respectively.It is to be noted that if we replace q by q n , then z 1 and α will be replaced by z n and β, respectively, where β has degree n over α.
In the next section, we give the values of r k,n , eta-function identities and modular equations, which will be used in our subsequent sections.
In the next three theorems, we state three eta-function identities of Yi [9].

Relations between H(q) and H(q n )
In this section, we state and prove some relations between H(q) and H(q n ).