We define a new parameter involving quotient of Ramanujan's function for positive real numbers and and study its several properties. We prove some general theorems for the explicit evaluations of the parameter and find many explicit values. Some values of are then used to find some new and known values of Ramanujan's class invariant .
1. Introduction
In Chapter 16 of his second notebook [1], Ramanujan develops the theory of theta-function. Ramanujan's general theta-function is defined by
After Ramanujan, for , we define
where . If with Im, then , where denotes the classical Dedekind eta function.
Ramanujan's function is defined by
The function is intimately connected to Ramanujan's class invariants and , which are defined by
where and is a positive rational number. Since from [2, page 124, Entry 12(v) & (vi)]
it follows from (4) that
Also, if has degree over , then
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 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 [6–12].
Ramanujan and Weber independently and many others in the literature calculated class invariants for odd values of and for even values of . For the first time, Yi [13] calculated some values of for odd values of by finding explicit values of the parameter (see [13, page 11, (2.1.1)] or [14, page 4, (1.11)]) defined by
In particular, she established the result [13, page 18, Theorem ]
However, the values of for even values of have not been calculated. The main objective of this paper is to evaluate some new values of for even values of . We also prove some known values of . For evaluation of class invariant in this paper, we introduce the parameter , which is defined as
where and are positive real numbers.
In Section 3, we study some properties of and also establish its relations with Ramanujan's class invariant . In Section 4, by employing Ramanujan's modular equations, we present some general theorems for the explicit evaluations of and find several explicit values of . In Section 5, we establish some general theorems connecting the parameter and the class invariant . We also evaluate some explicit values of the product by employing some values of evaluated in Section 4. Finally, in Section 6, we calculate new and known values of class invariant by combining the explicit values of and the product 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 and , we complete this introduction by defining Ramanujan's modular equation from Berndt's book [2]. The complete elliptic integral of the first kind is defined by
where , denotes the ordinary or Gaussian hypergeometric function, and
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 (13).
If we set
we see that (13) 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
Ramanujan also established many “mixed” modular equations in which four distinct moduli appear, which we define from Berndt's book [2, page 325].
Let , , , , , , , and denote complete elliptic integrals of the first kind corresponding, in pairs, to the moduli , , , and and their complementary moduli, respectively. Let , , and be positive integers such that . Suppose that the equalities
hold. Then, a “mixed” modular equation is a relation between the moduli , , , and that is induced by (16). In such an instance, we say that , , and are of degrees , , and , respectively, over or , , , and have degrees 1, , , and , respectively. Denoting , where
the multipliers and associated with , , and , , respectively, are defined by and .
2. Preliminary Results
Lemma 1 (see [2, page 43, Entry 27(v)]). If and are such that the modulus of each exponential argument is less than 1 and , then
Lemma 2 (see [15, page 241, Lemma 2.3]). Let and ; then
Lemma 3 (see [15, page 241, Lemma 2.8]). Let and ; then
Lemma 4 (see [15, page 252, Lemma 2.13]). Let and ; then
Lemma 5 (see [2, page 231, Entry 5(xii)]). Let and ; then
where has degree 3 over .
Lemma 6 (see [2, page 282, Entry 13(xiv)]). Let and ; then
where has degree 5 over .
Lemma 7 (see [2, page 315, Entry 13(xiv)]). Let and ; then
where has degree 7 over .
Lemma 8 (see [5, page 378, Entry 41]). Let and ; then
where has degree 13 over .
Lemma 9 (see [5, page 381, Entry 50]). If , , , and have degrees 1, 5, 7, and 35, respectively, then
Lemma 10 (see [5, page 381, Entry 51]). If , and have degrees 1, 13, 3, and 39, respectively, then
Lemma 11 (see [5, page 381, Entry 52]). If , and have degrees 1, 13, 5, and 65, respectively, then
Lemma 12 (see [16, page 277, Lemma 3.1]). If , and have degrees 1, 3, 7, and 21, respectively, then
Lemma 13 (see [15, page 243, Theorem 2.5]). If , , , and have degrees 1, 2, 3, and 6, respectively, then
Lemma 14 (see [15, page 248, Theorem 2.10]). If , , and have degrees 1, 2, 5, and 10, respectively, then
Lemma 15 (see [15, page 252, Theorem 2.12]). If , , and have degrees 1, 2, 7, and 14, respectively, then
3. Properties of
In this section, we study some properties of . We also establish a relation connecting and Ramanujan's class invariants .
Theorem 16. For all positive real numbers and , one has
Proof . Using the definition of and Lemma 1, we easily arrive at (i). Replacing by in and using Lemma 1, we find that , which completes the proof of (ii). To prove (iii), we use Lemma 1 in the definition of to arrive at ()1.
Remark 17. By using the definitions of and , it can be seen that has positive real value less than 1 and that the values of decrease as increases when . Thus, by Theorem 16(i), for all if .
Theorem 18. For all positive real numbers k, m, and n, one has
Proof. Using the definition of , 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
Proof. Setting in Theorem 18 and simplifying using Theorem 16(ii), we obtain
Replacing by , 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 , we deduce that, for positive real numbers , and ,
Now the result follows readily from (40), and the hypothesis that .
Corollary 21. For any positive real numbers and , we have
Proof. The result follows immediately from Theorem 20 with , and .
Now, we give some relations connecting the parameter and Ramanujan's class invariants .
Theorem 22. Let and be any positive real numbers. Then
Proof. Proof of (i) follows easily from the definitions of and from (10) and (4), respectively. To prove (ii), we set in part (i) and use Theorem 16(i) and (iii).
4. General Theorems and Explicit Evaluations of
In this section, we prove some general theorems for the explicit evaluations of and find its explicit values.
Theorem 23. One has
Proof. The proof follows easily from the definition of and Lemma 2.
Corollary 24. One has
Proof. Setting in Theorem 23 and using Theorem 16(ii), we obtain
Equivalently,
where
Solving (46) and using the fact in Remark 17, we obtain
Employing (48) in (47), solving the resulting equation for , and noting that , we arrive at
This completes the proof of (i). Again setting 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 , we obtain
This completes the proof of (ii). Now (iii) and (iv) follow from (i) and (ii), respectively, and Theorem 16(ii).
Theorem 25. One has
Proof. The proof follows from Lemma 3 and the definition of .
Corollary 26. One has
Proof. Setting in Theorem 25 and using Theorem 16(ii), we obtain
where
Solving (57) and noting the fact in Remark 17, we obtain
Employing (59) in (58), solving the resulting equation, and noting that , we obtain
This completes the proof of (i). Again, setting 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 , we arrive at
This completes the proof of (ii). Now (iii) and (iv) follow from (i) and (ii), respectively, and Theorem 16(ii).
Theorem 27. One has
Corollary 28. One has
Proof. Setting and simplifying using Theorem 16(ii), we obtain
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 , we obtain
This completes the proof of (i).
To prove (ii), setting 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 , we deduce that
So the proof of (ii) is complete.
Now (iii) and (iv) follow from (i) and (ii), respectively, and Theorem 16(ii).
Theorem 29. One has
Proof. Using (5) in Lemma 9, we find that
Setting and using the definition of in (77), we get
Employing (78) in (27), we complete the proof.
Corollary 30. One has
where and .
Proof. Setting in Theorem 29 and simplifying using Theorem 16(ii), we obtain
Equivalently,
where
Solving (81) and noting the fact in Remark 17, we obtain
Combining (82) and (83) and noting that , we deduce that
where
This completes the proof of (i). Again setting and simplifying using Theorem 16(i), we arrive at
where
Solving (86) and using Remark 17, we get
Combining (87) and (88) and noting that , we obtain
where
This completes the proof of (ii). Now (ii) and (iv) easily follow from (i) and (ii), respectively, and Theorem 16(ii).
Theorem 31. One has
Proof. Proceeding as in the proof of Theorem 29, using (5) in Lemma 10, setting , and using the definition of , we arrive at
Employing (92) in (28), we complete the proof.
Corollary 32. One has
Proof. Setting 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 , we obtain
So we complete the proof of (i). Again setting and using Theorem 16(i), we obtain
where
Solving (98) and using Remark 17, we get
Combing (99) and (100) and noting that , 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).
Theorem 33. One has
Proof. Using (5) in Lemma 11, setting , and using the definition of , we arrive at
Employing (103) in (29), we complete the proof.
Corollary 34. One has
where .
Proof. Setting and simplifying using Theorem 16(ii), we arrive at
where
Solving (105) and using the fact in Remark 17, we obtain
Employing (107) in (106), solving the resulting equation, and noting that , we obtain
This completes the proof of (i). To prove (ii), setting and simplifying using Theorem 16(i), we arrive at
where
Solving (109) and using the fact in Remark 17, we obtain
Employing (111) and (110), solving the resulting equation, and noting that , we obtain
where . 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
Proof. Using (5) in Lemma 12, setting , and using the definition of , we arrive at
Employing (114) in (30), we complete the proof.
Corollary 36. One has
Proof. Setting 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 and simplifying using Theorem 16(i), we arrive at
where
Solving (118) and using the fact in Remark 17, we obtain
Employing (120) in (119), solving the resulting equation, and noting that , 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).
5. General Theorems and Explicit Evaluations of
In this section we evaluate some explicit values of the product by establishing some general theorems and employing the values of obtained in Section 4. We recall from Theorem 22(ii) that for ready references in this section.
Theorem 37. One has
Proof. To prove (i), using (5) in Lemma 13, setting , and employing the definitions of and , we obtain
Employing (123) in (31), we complete the proof. (ii) follows similarly from Lemma 5 and the definition of and with .
Corollary 38. One has
Proof. Setting in Theorem 37(i) and simplifying using Theorem 16(ii) and the result , we obtain
Solving (125) and noting that , we complete the proof of (i). To prove (ii), setting in Theorem 37(i); using Theorem 16(i), and noting that , we obtain
Employing (53) in (126), solving the resulting equation, and noting that , we obtain
Using the value from [5, p. 189] in (127), we complete the proof of (ii). To prove (iii), setting in Theorem 37(ii), we obtain
Cubing (96) and then employing in (128) and solving the resulting equation, we complete the proof.
Theorem 39. One has
Proof. Using (5) in Lemma 14, setting , and employing the definitions of and , we obtain
Employing (130) in (32), we complete the proof of (i). Similarly, (ii) follows from Lemma 6 and the definition of and with .
Corollary 40. One has
where denotes .
Proof. Setting in Theorem 39(i) and simplifying using Theorem 16(ii) and the result
