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.
Throughout the paper, we assume . As usual, for positive integers and any complex number , we write
Ramanujan's general theta-function is defined by where . After Ramanujan, we define Ramanujan recorded many -continued fractions and some of their explicit values in his second notebook  and in his lost notebook . 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 , , and are complex numbers with , or , , and are complex numbers with for some integer . Several elegant -continued fractions have representations as -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  (also see ). The most famous one, of course, is the Rogers-Ramanujan continued fraction defined by The continued fraction has a very beautiful and extensive theory almost all of which was developed by Ramanujan. In particular, his lost notebook  contains several results on the Rogers-Ramanujan continued fraction. We refer to the paper by Berndt et al. , Kang [7, 8] for proofs of many of these theorems.
We complete this introduction by defining Ramanujan’s modular equation from Berndt’s book . The complete elliptic integral of the first kind is defined by where , denotes the ordinary or Gaussian hypergeometric function. 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 the moduli , , , and , respectively. Suppose that the equality holds for some positive integer . Then, a modular equation of degree is a relation between the moduli and which is implied by (1.11). If we set we see that (1.11) is equivalent to the relation . Thus, a modular equation can be viewed as an identity involving theta-functions at the arguments and . Ramanujan recorded his modular equations in terms of and , where and . We say that has degree over . The multiplier connecting and is defined by where .
2. 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
Lemma 2.2 (see [3, p. 214, Entry 24(iii)]). If has degree 2 over , then
Lemma 2.3 (see [3, p. 230, Entry 5(ii)]). If has degree 3 over , then
Lemma 2.4 (see [3, p. 215, (24.22)]). If has degree 4 over , then
Lemma 2.5 (see [3, p. 280-281, Entry 13(v) and (vi)]). If has degree 5 over , then
Lemma 2.6 (see [3, p. 314, Entry 19(i)]). If has degree 7 over , then
3. Modular Identitites for
In this section, we use Ramanujan's modular equations to prove certain modular identities for .
Theorem 3.1. One has
Corollary 3.2. One has
Proof. Dividing numerator and denominator on right-hand side of the identity in Theorem 3.1 by and simplifying, we complete the proof.
Theorem 3.3. One has where has degree n over .
Theorem 3.4. Let and . Then,
Theorem 3.5. Let and . Then,
Proof. Eliminating in (2.2) and then simplifying, we deduce that From Theorem 3.3(i), we have Now, employing Theorem 3.3(ii) with and (3.9) in (3.8) and factorizing using Mathematica, we obtain It can be seen that the first and the last factors in (3.10) do not vanish for . So, by identity theorem, we have
Theorem 3.6. Let and . Then,
Proof. From Lemma 2.3, we obtain
From Theorem 3.3, we deduce that
where has degree 3 over .
Employing (3.14) in (3.13) and factorizing using Mathematica, we arrive at It can be seen that the second factor of (3.15) does not vanish for , so by identity theorem, we have
Theorem 3.7. Let and . Then,
Proof. Squaring the modular equation in Lemma 2.4 and simplifying, we obtain From Theorem 3.3(i), we have Now, employing Theorem 3.3(ii) with and (3.19) in (3.18) and simplifying, we complete the proof.
Theorem 3.8. Let and . Then,
Proof. From Theorem 3.3, we obtain
where has degree 5 over .
Employing (3.21) in (2.5), we find that respectively.
Eliminating between (3.22) and (3.23) and simplifying, we deduce that Substituting for and from (3.21) in (3.24) and simplifying, we arrive at
Theorem 3.9. Let and . Then,
4. Explicit Evaluations of
In this section, we establish some general theorems for the explicit evaluations of the continued fraction and give examples.
For , Ramanujan's two class invariants and are defined by The class invariants and are connected by the relation [4, p. 187, Entry 2.1]:
The singular modulus is defined by , where is a positive integer and unique positive number between 0 and 1 satisfying
Class invariants and singular moduli are intimately related by the equalities [4, p. 185, ]: An account of Ramanujan's class invariants and singular moduli can be found in Chapter 34 of Berndt's book .
Theorem 4.1. One has
Proof. We set in Theorem 3.3(i) and use the definition of singular moduli and simplifying, we complete the proof.
In the scattered places of his first notebook , Ramanujan calculated over 30 singular moduli . See Chapter 34 of Berndt's book  for details. Thus, one can use Theorem 4.1 to find the values of if the corresponding values of are known. For example, from [4, p. 281, Theorem 9.2], we note that Employing (4.6) in Theorem 4.1, we calculate Many other values of can be computed by using the known values of .
Theorem 4.2. One has
Proof. Dividing numerator and denominator of right-hand side of Theorem 3.1 and employing (1.6), we obtain Setting , employing the definitions of and from (4.1) in (4.9) and simplifying, we obtain Substituting for from (4.2) in (4.10) and simplifying, we complete the proof.
Theorem 4.2 implies that if we know the values of and for any positive number , then corresponding values of can easily be calculated. Saikia  evaluated several values of and for positive number . For example, noting from [9, Theorem 3.5], we have Employing (4.11) in Theorem 4.2, we obtain Many other values of can be determined by using the values of and evaluated in .
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B. C. Berndt, “Flowers which we cannot yet see growing in Ramanujan's garden of hypergeometric series, elliptic functions, and 's,” in Special Functions 2000: Current Perspective and Future Directions, J. Bustoz, M. E. H. Ismail, and S. K. Suslov, Eds., vol. 30, pp. 61–85, Kluwer Academic, Dordrecht, The Netherlands, 2001.View at: Publisher Site | Google Scholar
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