Modular Identities and Explicit Evaluations of a Continued Fraction of Ramanujan

We study a new continued fraction of Ramanujan. We prove its modular identities and give some explicit evaluations.


Introduction
Throughout the paper, we assume |q| < 1. As usual, for positive integers n and any complex number a, we write a n : a; q n : where |ab| < 1. After Ramanujan, we define Ramanujan recorded many q-continued fractions and some of their explicit values in his second notebook 1 and in his lost notebook 2 . The following beautiful continued fraction identity was recorded by Ramanujan in his second notebook and can be found in 3, p. 11, Entry 11 : where either q, a, and b are complex numbers with |q| < 1, or q, a, and b are complex numbers with a bq m for some integer m. Several elegant q-continued fractions have representations as q-products and some of them can be expressed in terms of Ramanujan's theta-functions. An account of this can be found in in Chapter 32 of Berndt's book 4 also see 5 . The most famous one, of course, is the Rogers-Ramanujan continued fraction R q defined by The continued fraction R q has a very beautiful and extensive theory almost all of which was developed by Ramanujan. In particular, his lost notebook 2 contains several results on the Rogers-Ramanujan continued fraction. We refer to the paper by Berndt et al. 6 , Kang 7,8 for proofs of many of these theorems. In this paper, we examine another continued fraction T q of Ramanujan arising from 1.7 and is defined by 1.9 Note that, replacing q by q 2 and then setting a q and b 0 in 1.7 , we obtain 1.9 .
In Section 2, we record some preliminary results. Section 3 is devoted to prove some modular identities for the continued fraction T q . Finally, in Section 4, we give some explicit evaluations of T q .

International Journal of Mathematics and Mathematical Sciences 3
We complete this introduction by defining Ramanujan's modular equation from Berndt's book 3 . The complete elliptic integral of the first kind K k is defined by where 0 < k < 1, 2 F 1 denotes the ordinary or Gaussian hypergeometric function. The number k is called the modulus of K, and k : √ 1 − k 2 is called the complementary modulus. Let K, K , L, and L denote the complete elliptic integrals of the first kind associated with the moduli k, k , l, and l , 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 1.11 . If we set we see that 1.11 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 where z r φ 2 q r .

Preliminary Results
In this section, we record some results that will be used in the subsequent sections.
Lemma 2.1 see 3, p. 124, Entry 12 i and ii . One has

Modular Identitites for T q
In this section, we use Ramanujan's modular equations to prove certain modular identities for T q .
Proof. Replacing q by q 2 and the setting a q and b 0 in 1.7 and simplifying, we obtain 3.2 Employing 1.6 and 1.9 in 3.2 and simplifying, we complete the proof. Proof. Dividing numerator and denominator on right-hand side of the identity in Theorem 3.1 by f −q and simplifying, we complete the proof.

Theorem 3.3. One has
where β has degree n over α.
Proof. We employ Lemma 2.1 in Corollary 3.2 to complete the proof. Proof. Replacing q by −q in Corollary 3.2, we obtain Now, eliminating f q /f −q between 3.6 and Corollary 3.2 and simplifying, we complete the proof.

Theorem 3.5. Let u T q and v T q 2 . Then,
Proof. Eliminating m in 2.2 and then simplifying, we deduce that

3.9
Now, employing Theorem 3.3 ii with n 2 and 3.9 in 3.8 and factorizing using Mathematica, we obtain 3.10 6

International Journal of Mathematics and Mathematical Sciences
It can be seen that the first and the last factors in 3.10 do not vanish for |q| → 0. So, by identity theorem, we have Theorem 3.6. Let u T q and v T q 3 . Then, Proof. From Lemma 2.3, we obtain From Theorem 3.3, we deduce that 3.14 where β has degree 3 over α. Employing 3.14 in 3.13 and factorizing using Mathematica, we arrive at

3.15
It can be seen that the second factor of 3.15 does not vanish for |q| → 0, so by identity theorem, we have Theorem 3.7. Let u T q and v T q 4 . Then,

3.19
Now, employing Theorem 3.3 ii with n 4 and 3.19 in 3.18 and simplifying, we complete the proof.
Theorem 3.8. Let u T q and v T q 5 . Then, Proof. From Theorem 3.3, we obtain where β has degree 5 over α. Employing 3.21 in 2.5 , we find that respectively. Eliminating m between 3.22 and 3.23 and simplifying, we deduce that Substituting for c and d from 3.21 in 3.24 and simplifying, we arrive at International Journal of Mathematics and Mathematical Sciences Theorem 3.9. Let u T q and v T q 7 . Then, Proof. From Lemma 2.6, we obtain Again, from Theorem 3.3, we deduce that

3.28
where β has degree 7 over α. Employing 3.28 in 3.27 and simplifying using Mathematica, we arrive at 3.29

Explicit Evaluations of T q
In this section, we establish some general theorems for the explicit evaluations of the continued fraction T q and give examples. For q : e −π √ n , Ramanujan's two class invariants G n and g n are defined by The class invariants G n and g n are connected by the relation 4, p. 187, Entry 2.1 : The singular modulus α n is defined by α n : α e −π √ n , where n is a positive integer and unique positive number between 0 and 1 satisfying √ Many other values of T e −π √ n can be computed by using the known values of α n .

4.8
Proof. Dividing numerator and denominator of right-hand side of Theorem 3.1 and employing 1.6 , we obtain T q χ q − χ −q χ q χ −q . 4.9