A Parameter for Ramanujan’s Function χ(q): Its Explicit Values and Applications

We define a new parameter 𝐼𝑘,𝑛 involving quotient of Ramanujan's function 𝜒(𝑞) for positive real numbers 𝑘 and 𝑛 and study its several properties. We prove some general theorems for the explicit evaluations of the parameter 𝐼𝑘,𝑛 and find many explicit values. Some values of 𝐼𝑘,𝑛 are then used to find some new and known values of Ramanujan's class invariant 𝐺𝑛.

In his notebooks [1] and paper [3], Ramanujan recorded a total of 116 class invariants or monic polynomials satisfied by them. The table at the end of Weber's book [4, page 721-726] also contains the values of 107 class invariants. 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, and so on. An account of Ramanujan's class invariants and applications can be found in Berndt's book [5]. For further references, see [6][7][8][9][10][11][12]. Ramanujan and Weber independently and many others in the literature calculated class invariants G n for odd values of n and g n for even values of n. For the first time, Yi [13] calculated some values of g n for odd values of n by finding explicit values of the parameter r k,n (see [13, page 11, (2.1.1)] or [14, page 4, (1.11)]) defined by r k,n := f −q k 1/4 q (k−1)/24 f −q k , q = e −2π √ n/k . (8) In particular, she established the result [13, page 18, Theorem 2.2.3] g n = r 2,n/2 .
However, the values of G n for even values of n have not been calculated. The main objective of this paper is to evaluate some new values of G n for even values of n. We also prove some known values of G n . For evaluation of class invariant G n in this paper, we introduce the parameter I k,n , which is defined as where k and n are positive real numbers.
In Section 3, we study some properties of I k,n and also establish its relations with Ramanujan's class invariant G n . In Section 4, by employing Ramanujan's modular equations, we present some general theorems for the explicit evaluations of I k,n and find several explicit values of I k,n . In Section 5, we establish some general theorems connecting the parameter I k,n and the class invariant G n . We also evaluate some explicit values of the product G nk G n/k by employing some values of I k,n evaluated in Section 4. Finally, in Section 6, we calculate new and known values of class invariant G n by combining the explicit values of I k,n and the product G nk G n/k evaluated in Sections 4 and 5, respectively. Section 2 is devoted to record some preliminary results.
Since Ramanujan's modular equations are key in our evaluations of I k,n and G n , we complete this introduction by defining Ramanujan's modular equation from Berndt's book [2]. 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, and (a) n = a(a + 1)(a + 2) · · · (a + n − 1).
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 (13). If we set we see that (13) 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 Ramanujan also established many "mixed" modular equations in which four distinct moduli appear, which we define from Berndt's book [2, page 325]. Let K, K , L 1 , L 1 , L 2 , L 2 , L 3 , and L 3 denote complete elliptic integrals of the first kind corresponding, in pairs, to the moduli √ α, β, √ γ, and √ δ and their complementary moduli, respectively. Let n 1 , n 2 , and n 3 be positive integers such that n 3 = n 1 n 2 . Suppose that the equalities hold. Then, a "mixed" modular equation is a relation between the moduli √ α, β, √ γ, and √ δ that is induced by (16). In such an instance, we say that β, γ, and δ are of degrees n 1 , n 2 , and n 3 , respectively, over α or α, β, γ, and δ have degrees 1, n 1 , n 2 , and n 3 , respectively. Denoting z r = φ 2 (q r ), where the multipliers m and m associated with α, β, and γ, δ, respectively, are defined by m = z 1 /z n1 and m = z n2 /z n3 .

Properties of I k,n
In this section, we study some properties of I k,n . We also establish a relation connecting I k,n and Ramanujan's class invariants G n .

Theorem 16.
For all positive real numbers k and n, one has Proof . Using the definition of I k,n and Lemma 1, we easily arrive at (i). Replacing n by 1/n in I k,n and using Lemma 1, we find that I k,n I k,1/n = 1, which completes the proof of (ii).
To prove (iii), we use Lemma 1 in the definition of I k,n to arrive at (I k,n /I n,k )=1.
Remark 17. By using the definitions of χ(q) and I k,n , it can be seen that I k,n has positive real value less than 1 and that the values of I k,n decrease as n increases when k > 1. Thus, by Theorem 16(i), I k,n < 1 for all n > 1 if k > 1.
Theorem 18. For all positive real numbers k, m, and n, one has Proof. Using the definition of I k,n , we obtain Using Lemma 1 in the denominator of the right-hand side of (36) and simplifying, we complete the proof.

Corollary 19. For all positive real numbers k and n, one has
I k 2 ,n = I nk,n I k,n/k .
Proof. Setting k = n in Theorem 18 and simplifying using Theorem 16(ii), we obtain Replacing m by n, we complete the proof.
Theorem 20. Let k, a, b, c, and d be positive real numbers such that ab=cd. Then Proof. From the definition of I k,n , we deduce that, for positive real numbers k, a, b, c, and d, Now the result follows readily from (40), and the hypothesis that ab = cd.
Corollary 21. For any positive real numbers n and p, we have Proof. The result follows immediately from Theorem 20 with a = p 2 , b = 1, c = d = p, and k = n. Now, we give some relations connecting the parameter I k,n and Ramanujan's class invariants G n .

Theorem 22. Let k and n be any positive real numbers. Then
Proof. Proof of (i) follows easily from the definitions of I k,n and G n from (10) and (4), respectively. To prove (ii), we set k = 1 in part (i) and use Theorem 16(i) and (iii).

General Theorems and Explicit Evaluations of I k,n
In this section, we prove some general theorems for the explicit evaluations of I k,n and find its explicit values. (43) Proof. The proof follows easily from the definition of I k,n and Lemma 2.
Corollary 24. One has Proof. Setting n = 1/2 in Theorem 23 and using Theorem 16(ii), we obtain I 24 3,2 + I −24 3,2 + 176 I 12 3,2 + I −12 Equivalently, where Solving (46) and using the fact in Remark 17, we obtain Employing (48) in (47), solving the resulting equation for I 3,2 , and noting that I 3,2 < 1, we arrive at This completes the proof of (i). Again setting n = 1 in Theorem 23 and using Theorem 16(i), we obtain Equivalently, where Since the first factor of (51) is nonzero, solving the second factor, we deduce that Employing (53) in (52), solving the resulting equation, and using the fact that I 3,4 < 1, we obtain This completes the proof of (ii). Now (iii) and (iv) follow from (i) and (ii), respectively, and Theorem 16(ii).
Proof. Setting n = 1/2 in Theorem 25 and using Theorem 16(ii), we obtain Solving (57) and noting the fact in Remark 17, we obtain Employing (59) in (58), solving the resulting equation, and noting that I 5,2 < 1, we obtain This completes the proof of (i). Again, setting n = 1 in Theorem 25 and using Theorem 16(i), we obtain where Solving (61), we obtain Using (63) in (62), solving the resulting equation, and noting that I 5,4 < 1, we arrive at This completes the proof of (ii). Now (iii) and (iv) follow from (i) and (ii), respectively, and Theorem 16(ii). Equivalently, where By using the fact in Remark 17, it is seen that the first factor of (68) is nonzero, and so from the second factor, we deduce that Combining (69) and (70) and noting that I 7,2 < 1, we obtain This completes the proof of (i).
To prove (ii), setting n = 1 and simplifying using Theorem 16(i), we arrive at where Using the fact in Remark 17 it is seen that the first two factors of (72) are nonzero, and so solving the third factor, we obtain Combining (73) and (74) and noting that I 7,4 < 1, we deduce that So the proof of (ii) is complete. Now (iii) and (iv) follow from (i) and (ii), respectively, and Theorem 16(ii).
This completes the proof of (i). Again setting n = 1 and simplifying using Theorem 16(i), we arrive at This completes the proof of (ii). Now (ii) and (iv) easily follow from (i) and (ii), respectively, and Theorem 16(ii). (93) Proof. Setting n = 1/3 in Theorem 31 and simplifying using Theorem 16(ii), we obtain where Solving (94) and using Remark 17, we get Combining (95) and (96) and noting that I 13,3 < 1, we obtain So we complete the proof of (i). Again setting n = 1 and using Theorem 16(i), we obtain where J = I 13,9 + I −1 13,9 .
Solving (98) and using Remark 17, we get Combing (99) and (100) and noting that I 13,9 < 1, we deduce that So the proofs of (ii) is complete. Now the proof of (iii) and (iv) follow from (i) and (ii), respectively, and Theorem 16(ii).
Solving (105) and using the fact in Remark 17, we obtain Employing (107) in (106), solving the resulting equation, and noting that I 13,5 < 1, we obtain This completes the proof of (i).
To prove (ii), setting n = 1 and simplifying using Theorem 16(i), we arrive at This completes the proof of (ii). Now the proofs of (iii) and (iv) follow from (i) and (ii), respectively, and Theorem 16(ii).
Theorem 35. One has I 7,n I 7,9n + I 7,n I 7,9n Proof. Setting n = 1/3 and simplifying using Theorem 16(ii), we arrive at Solving (116) and noting the fact in Remark 17, we obtain This completes the proof of (i).
To prove (ii), setting n = 1 and simplifying using Theorem 16(i), we arrive at where

ISRN Computational Mathematics
Solving (118) and using the fact in Remark 17, we obtain Employing (120) in (119), solving the resulting equation, and noting that I 7,9 < 1, we deduce that This completes the proof of (ii). Now the proofs of (iii) and (iv) follow from (i) and (ii), respectively, and Theorem 16(ii).

General Theorems and Explicit Evaluations of G nk G n/k
In this section we evaluate some explicit values of the product G nk G n/k by establishing some general theorems and employing the values of I k,n obtained in Section 4. We recall from Theorem 22(ii) that G 1/n = G n for ready references in this section.

Corollary 42. One has
Proof. Setting n = 1/7 and simplifying using Theorem 16(ii) and the result G 1/n = G n , we get Solving (138) and noting that G 14 G 7/2 > 1, we complete the proof.
To prove (iii), setting n = 9 in Theorem 43, we get Proof. Using (5) in Lemma 8, setting q := e −π √ n/13 , and employing definitions of I k,n and G n , we arrive at Solving the resulting equation (151) and noting that G 9/13 G 117 > 1, we complete the proof.

New Values of Class Invariant G n
In this section we find some new values of Ramanujan's class invariant G n by using explicit values of I k,n and G nk G n/k evaluated in Sections 3 and 4, respectively. For ready references in this section, we recall from Theorem 22 that I k,n = G n/k /G nk . We also recall from Theorem 22(ii) that G 1/n = G n .
The proofs of the Theorems 48-56 are identical to the proof of Theorem 47. So we give the references of the required results only.
Proof. We employ Corollary 26(ii) and Corollary 40(ii) and proceed as in Theorem 47.