Ramanujan’s modular equations and Weber–Ramanujan class invariants Gn and gn

In this paper, we use Ramanujan’s modular equations and transformation formulas to find several new values of Weber–Ramanujan class invariants gn. We also give new approach to some known values of class invariants Gn.

. (1.3) In his notebooks [11] and paper [10], Ramanujan recorded a total of 116 class invariants. The table at the end of Weber's book [12, p. 721-726] contains the values of 107 class invariants. There are many applications of Weber-Ramanujan class invariants G n and g n . Weber primarily was motivated to calculate class invariants so that he could construct Hilbert class fields. On the other hand Ramanujan calculated class invariants to approximate π , and probably for finding explicit values of Rogers-Ramanujan continued fractions, theta-functions, etc. For further applications of class invariants see [5][6][7][8][9]. An account of Ramanujan's class invariants and applications can also be found in Berndt's book [4].
In 2001, Yi [13] evaluated several new class invariants G n and g n by finding explicit values of the parameter r k,n (see [13, p. 11 Adiga et al. [1] and Baruah [2] also evaluated some new values of G n and g n . In this paper, we find further new values of Ramanujan's class invariant g n and also give new approach to some known values of G n . We consider the parameter A n defined by A n = f (−q) 2 1/4 q 1/24 f (−q 2 ) , q := e −2π √ n/2 (1.6) where n is a positive rational number. Clearly, the parameter A n is the particular case, k = 2, of the Yi's parameter r k,n defined above. In Sect. 2, we record some transformation formulas and modular equations. We also prove four eta-function identities which are also particular type of modular equations. In Sect. 3, we find several values of Weber-Ramanujan class invariants g n . Finally in Sect. 4, we use some new values of g n evaluated in Sect. 3 to prove some known values on G n .
Since modular equations are key in our evaluations, we now define a modular equation. 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 (1.7). 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 multipliers m associated with α, β is defined by m = z 1 /z n .

Transformation formulas and modular equations
The section is devoted to record some transformation formulas and modular equations which will be used in next section. The eta-function identities in Lemmas 2.7-2.9 are found by Yi [13] by using Garvan's Maple q-series package and then proved by verification. We give direct proofs of these identities. In our proofs, it is not necessary to know the identities in advance and employ Ramanujan's modular equations and transformation formulas. Eta-function identity in Lemma 2.10 is new.
then, if β has degree 9 over α, then, if β has degree 19 over α, Proof Transcribing using Lemma 2.2, we get Proof Transcribing P and Q by using Lemma 2.2 and simplifying, we get α = 16 16 + P 8 and β = where β has degree 3 over α. It follows that (2.21) Using (2.20) and (2.21) in Lemma 2.3, we arrive at Since the second factor is non-zero in a neighborhood of the origin, we deduce Dividing above equation by P 2 Q 2 , we complete the proof.
Proof By using Lemma 2.2, we find that α = 16 16 + P 8 and β = where β has degree 5 over α. It follows that Now, combining (2.6) and (2.7), we obtain (2.27) Employing (2.25) and (2.26) in (2.27) and simplifying, we obtain Factorizing (2.28) using Mathematica, we obtain Now, proceeding as in the proof of Lemma 2.13, it can be shown that the first and last factors of (2.29) are non-zero in a neighborhood of zero. Thus, we have Dividing above equation by P 3 Q 3 and rearranging the terms, we complete the proof.
Proof Proceeding as in the proof of Lemma 2.13, we obtain where β has degree 7 over α. Employing (2.31) in Lemma 2.5 and simplifying, we deduce that Simplifying (2.32), we get Dividing above equation by P 4 Q 4 and rearranging the terms, we complete the proof.

Evaluation of class invariants g n
Proof To prove (i) we use the definition of A n and Lemma 1.4. We set n = 1 in (i) to prove (ii). (iii) follows directly from the definitions of A n and g n from (1.6) and (1.2), respectively.

Theorem 3.2 One has
where β has degree k over α.
Proof For positive number k, set and Proof follows easily from Lemma 2.12 and the definition of A n .

Theorem 3.4 We have
Proof We set q := e −2π √ n/2 in Lemma 2.13 and use the definition of A n .

Theorem 3.5 We have
Proof Setting n = 1/6 in Theorem 3.4 and simplifying using Theorem 3.1(i), we obtain Now setting n = 1/6 in Theorem 3.3 and simplifying by using Theorem 3.1(i), we obtain Using (3.3) in (3.5), we deduce that Theorem 3. 6 We have Proof We set q := e −2π √ n/2 in Lemma 2.14 and use the definition of A n .

Theorem 3.8 We have
Employing (3.20) in (2.9) and factorizing, we obtain The first factor in (3.22) is not zero and do not give real value of x such the 0 < x < 1, so we have Solving (3.23) and noting 0 < x < 1 and (A 18 A 9/2 ) 4 > 1 is real, we get Since first two factors are not zero, so we have

Evaluation of class invariants G n
In his paper [10] and also in page 294 of his second notebook [11, Vol. II], Ramanujan recorded two simple formulas relating the class invariants g n and G n , namely, for n > 0 g 4n = 2 1/4 g n G n (4.1) and (g n G n ) 8 (G 8 n − g 8 n ) = Thus, if we know g n and g 4n or only g n then the corresponding G n can be calculated by the above formulas. We now find some values of G n by using (4.1) and the new values of g n and g 4n evaluated in above section. , (vii) G 19 = 2 5/12 3 1/6 r −1/12 , where r is given in Theorem 3.13.
Proof For (i), we use the values of g 12 and g 3 from Theorem 3.5 in (4.1). To prove (ii), we employ the values of g 20 and g 5 from Theorem 3.7 in (4.1). To prove (iii), employ the values of g 28 and g 7 from Theorem 3.9 in (4.1). To prove (iv), we use the values of g 36 and g 9 from Theorem 3.10 in (4.1). Employing the values of g 52 and g 13 from Theorem 3.11 in (4.1) we arrive at (v). Employing the values of g 17 and g 68 from Theorem 3.12 in (4.1), we prove (vi). For (vii), we use the values of g 19 and g 76 from Theorem 3.13 in (4.1). To prove (viii), we use the values of g 27 and g 108 from Theorem 3.14 in (4.1). To prove (ix), we employ the values of g 15 and g 60 from Theorem 3.15 in (4.1).