ISRN Computational MathematicsVolume 2012 (2012), Article ID 169050, 14 pagesdoi:10.5402/2012/169050

Research Article

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

Nipen Saikia

Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh 791112, India

Received 3 May 2012; Accepted 28 June 2012

Academic Editors: L. Hajdu, L. S. Heath, and H. J. Ruskin

Copyright © 2012 Nipen Saikia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 .

For Lemmas 9 to 15, we set

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 , we obtain Solving (132) and noting that , we complete the proof of (i).For proof of (ii), setting in Theorem 39(ii) and simplifying using Theorem 18(i) and the result , we obtain Employing the value of from (63) in (133), solving the resulting equation, and noting that , we get So the proof is complete.

Theorem 41. One has

Proof. Using (5) in Lemma 15, setting , and employing the definitions of and , we obtain, Employing (136) in (33), we complete the proof.

Corollary 42. One has

Proof. Setting and simplifying using Theorem 16(ii) and the result , we get Solving (138) and noting that , we complete the proof.

Theorem 43. One has

Proof. Using (5) in Lemma 7, setting , and employing definitions of and , we arrive at Using (140) in (24), we complete the proof.

Corollary 44. One has

Proof. Setting in Theorem 43, we get Squaring (74) and simplifying, we obtain Employing (143) in (142), solving the resulting equation, and noting that , we complete the proof of (i).To prove (ii), we set in Theorem 43, and employing (116), we obatin Solving (144) and noting that , we complete the proof.To prove (iii), setting in Theorem 43, we get Squaring (120) twice and simplifying, we obtain Employing (146) in (145), solving the resulting equation, and noting that , we complete the proof.

Theorem 45. One has

Proof. Using (5) in Lemma 8, setting , and employing definitions of and , we arrive at Using (148) in (25), we complete the proof.

Corollary 46. One has

Proof. Setting in Theorem 45, we get Employing (100) in (150) and simplifying, we obtain Solving the resulting equation (151) and noting that , we complete the proof.

6. New Values of Class Invariant

In this section we find some new values of Ramanujan's class invariant by using explicit values of and evaluated in Sections 3 and 4, respectively. For ready references in this section, we recall from Theorem 22 that . We also recall from Theorem 22(ii) that .

Theorem 47. One has

Proof. From Corollary 24(i), we have Also, from Corollary 38(i), we have Multiplying (153) and (154) and simplifying, we complete the proof of (i). Dividing (154) by (153) and simplifying, we complete the proof of (ii).

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.

Theorem 48. One has

Proof. We use Corollary 24(ii) and Corollary 38(ii) and proceed as in Theorem 47.

Theorem 49. One has

Proof. We employ Corollary 26(i) and Corollary 40(i) and proceed as in Theorem 47.

Theorem 50. One has

Proof. We employ Corollary 26(ii) and Corollary 40(ii) and proceed as in Theorem 47.

Theorem 51. One has

Proof. We employ Corollary 28(i) and Corollary 42(i) and proceed as in Theorem 47.

Theorem 52. One has

Proof. We employ Corollary 28(ii) and Corollary 44(i) and proceed as in Theorem 47.

Theorem 53. One has The values and can also be found in [5, 6].

Proof. We employ Corollary 36(i) and Corollary 44(ii) and proceed as in Theorem 47.

Theorem 54. One has The value can also be found in [5, page 192].

Proof. We employ Corollary 36(ii) and Corollary 44(ii) and proceed as in Theorem 47.

Theorem 55. One has The values and can also be found in [5, 6].

Proof. We use Corollary 32(i) and Corollary 38(iii) and proceed as in Theorem 47.

Theorem 56. One has The values and can also be found in [5, page 193].

Proof. We employ Corollary 32(ii) and Corollary 46 and proceed as in Theorem 47.

References

S. Ramanujan, Notebooks, vol. 1-2, Tata Institute of Fundamental Research, Bombay, India, 1957.

B. C. Berndt, Ramanujan's Notebooks. Part III, Springer, New York, NY, USA, 1991.

S. Ramanujan, “Modular equations and approximations to π,” Quarterly Journal of Mathematics, vol. 45, pp. 350–372, 1914.

H. Weber, Lehrburg Der Algebra II,, Chelsea, New York, NY, USA, 1961.

B. C. Berndt, Ramanujan's Notebooks. Part V, Springer, New York, NY, USA, 1998.

N. D. Baruah, “On some class invariants of Ramanujan,” The Journal of the Indian Mathematical Society, vol. 68, no. 1–4, pp. 113–131, 2001.

B. C. Berndt and H. H. Chan, “Some values for the Rogers-Ramanujan continued fraction,” Canadian Journal of Mathematics, vol. 47, no. 5, pp. 897–914, 1995.

B. C. Berndt, H. H. Chan, S. Y. Kang, and L. C. Zhang, “A certain quotient of eta-functions found in Ramanujan's lost notebook,” Pacific Journal of Mathematics, vol. 202, no. 2, pp. 267–304, 2002.

B. C. Berndt, H. H. Chan, and L.-C. Zhang, “Ramanujan's class invariants with applications to the values of -continued fractions and theta functions,” in Special Functions, q-Series and Related Topics, M. Ismail, D. Masson, and M. Rahman, Eds., vol. 14 of Fields Institute Communications Series, pp. 37–53, American Mathematical Society, Providence, RI, USA, 1997.

B. C. Berndt, H. H. Chan, and L. C. Zhang, “Ramanujan's class invariants and cubic continued fraction,” Acta Arithmetica, vol. 73, no. 1, pp. 67–85, 1995.

B. C. Berndt, H. H. Chan, and L. C. Zhang, “Ramanujan's remarkable product of theta-functions,” Proceedings of the Edinburgh Mathematical Society, vol. 40, no. 3, pp. 583–612, 1997.

N. Saikia, “Ramanujan's modular equations and Weber-Ramanujan's class invariants Gn and gn,” Bulletin of Mathematical Sciences, vol. 2, no. 1, pp. 205–223, 2012.

J. Yi, Construction and application of modular equation [Ph.D. thesis], University of Illionis, 2001.

J. Yi, “Theta-function identities and the explicity formulas for theta-function and their applications,” Journal of Mathematical Analysis and Applications, vol. 292, no. 2, pp. 381–400, 2004.

K. R. Vasuki and T. G. Sreeramamurthy, “Certain new Ramanujan's Schläfli-type mixed modular equations,” Journal of Mathematical Analysis and Applications, vol. 309, no. 1, pp. 238–255, 2005.

N. D. Baruah, “On some of Ramanujan's Schläfli-type 'mixed' modular equations,” Journal of Number Theory, vol. 100, no. 2, pp. 270–294, 2003.