International Scholarly Research Notices

International Scholarly Research Notices / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 435261 | 8 pages | https://doi.org/10.1155/2013/435261

Some New Explicit Values of Parameters and of Quotients of Eta-Function

Academic Editor: T. Allahviranloo
Received09 Nov 2012
Accepted16 Dec 2012
Published18 Feb 2013

Abstract

We find some new explicit values of the parameters and of quotients of eta-function by using Ramanujan's class invariants.

1. Introduction

For and , the Dedekind eta-function and Ramanujan’s function are defined by where .

In page 212 of his lost notebook [1], Ramanujan defined and provided a list of eleven recorded and ten unrecorded values of for positive integers . All 21 values of and many more were established by Berndt et al. [2] by using the modular -invariant, modular equations, Kronecker’s limit formula, and the explicit Shimura reciprocity law. An account of this can also be found in Chapter 9 of [3]. Closely related to is the parameter introduced by Ramanathan [4] and is defined as Yi [5, 6] also found several values of parameters and by finding the explicit values of her parameters and , defined by where and are positive real numbers. In fact, , and . Baruah and Saikia [7, 8] also evaluated several new values of and . A generalization of has been studied by Naika et al. [9]. Here we give a list of the values of for which and were evaluated in [1, 2, 58] without referring them specifically to avoid repetitions. The values of for which were evaluated in the literature are = 1, 3, 5, 7, 9, 11, 13, 15, 5/3, 17, 25, 33, 41, 49, 57, 65, 81, 89, 73, 97, 121, 169, 193, 217, 241, 265, 289, 361, 11/3, and 19/3, and the values of values of for which were evaluated in literature are = 1, 2, 3, 4, 5, 6, 3/2, 7, 8, 9, 10, 5/2, 11, 13, 14, 7/2, 15, 5/3, 17, 19, 20, 5/4, 22, 11/2, 25, 26, 13/2, 44, 11/4, and 49. We also note from [2, page 281] and [7, page 43, Theorem 4.2] that respectively.

In this paper we find further new values of and by using Ramanujan’s class invariants which are defined as where . In particular, we evaluate values of for = 23, 31, 47, 59, 71, 2, 4, 6, 10, 5/2, 14, and 7/2 and new values of for = 23, 31, 47, 59, and 71. It worth to mention here that for the first time in this paper explicit values of for some even values of are evaluated. Previously, Berndt et al. [2] calculated explicit values of , , and by using Ramanujan’s class invariants. An account of Ramanujan’s class invariants can be found in Berndt’s book [10]. For further references on Ramanujan’s class invariants refer see [1114].

The explicit values of the functions and can be applied to find explicit values of Ramanujan’s cubic continued fraction defined by We refer to [7] for details. To this end we define a general parameter [14, page 2, equation (10)] for all positive real numbers and and connected with parameters , , and Ramanujan’s class invariants as Saikia [14] studied several properties of the parameter and evaluated many explicit values of by using Ramanujan’s modular equations. From [14, page 4, Theorem 16(iii)], we also note here that .

In Section 2, we record some preliminary results, and in Section 3 we evaluate new values of the parameters and .

2. Preliminaries

Lemma 1 (see [3, page 203, Theorem 9.3.1]). One has

Lemma 2 (see [14, page 4, Theorem 22(i)]). One has

Lemma 3. One has

Proof. We set in Lemma 2 and apply Lemma 1 to complete the proof.

Lemma 4. One has

This readily follows from [2, page 9, equation (3.4)] or [3, page 203, equation (9.3.4)].

Lemma 5. One has

Proof. From [11, page 123, equation (4.14)], we note that Replacing by in (14) and squaring, we arrive at desired result.

Lemma 6. One has

For (15), (16), (17) and (18) see [10, page 342]. For (19), (20), and (21) see [10, page 227, equation (5.8)], [10, page 229, equation (5.16)], and [10, page 257, equation (6.61)]. For (22) and (23) see [14, page 5, Corollary 24(i & ii))].

Lemma 7. One has

For (24) to (27) see [12]. Equation (28) follows from the values of and from [10, page 200].

Lemma 8. One has

Proof. From [15, page 124], we note that Employing (30) in (6) and simplifying, we deduce that From [15, page 230, Entry 5(ii)], we note that where has degree 3 over .
Combining (31) and (32) and simplifying, we deduce that From [10, page 187, Entry 2.2], we note that Employing (34) in (33), and simplifying, we find that Solving (35) for , we arrive at the desired result.

Lemma 9. One has

Proof. We set = 10/3, 2/15, 14/3, 6/7, and 6/3 in Lemma 8 and employ the corresponding values of from Lemma 7 to complete the proof.

3. New Values of and

The section is devoted to evaluate new values of and .

Theorem 10. One has

Proof. Employing (19) in Lemma 3 with , we have From (42), we deduce that Setting in Lemma 4, employing (43), and solving the resulting equation, we find that Multiplying (42) and (44) and simplifying, we easily calculate the value of , and dividing (44) by (42) and simplifying we evaluate the value of .

Remark 11. The values of and also follow from (5) and the values of and , respectively.

Theorem 12. One has

Proof. Employing (20) in Lemma 3 with , we have From (46), we deduce that Setting in Lemma 4, employing (47), and solving the resulting equation, we obtain Multiplying (46) and (48) and simplifying, we easily calculate the value of , and dividing (48) by (46) and simplifying we evaluate the value of .

Remark 13. The values of and also follow from (5) and the values of and , respectively.

Theorem 14. One has

Proof. Employing (21) in Lemma 3 with , we have From (50), we deduce that Setting in Lemma 4, employing (51), and solving the resulting equation, we obtain Combining (50) and (52) we easily evaluate the values of and .

Remark 15. The values of and also follow from (5) and the values of and , respectively.

Theorem 16. One has

Proof. Combining (15), (16) and Lemma 3, we deduce that From (54), we find that Setting in Lemma 4, employing (55), and solving the resulting equation, we obtain Combining (54) and (56) we easily evaluate the values of and .

Remark 17. The values of and can easily be calculated by employing the values of and , respectively, in (5).

Theorem 18. One has

Proof. Combining (17), (18) and Lemma 3, we deduce that From (58), we find that Setting in Lemma 4, employing (59), and solving the resulting equation, we obtain Combining (58) and (60) we easily calculate the values of and .

Remark 19. The values of and can easily be calculated by employing the values of and , respectively, in (5).

Next, we evaluate some values of and for even values of . The values of are new. The values of are classical, but the method of evaluations is new.

Theorem 20. One has

Proof. Combining (22) and Lemma 3 with , we note that From (62), we deduce that Setting in Lemma 4, employing (63), and solving the resulting equation, we obtain Combining (62) and (64) we easily evaluate the values of and .

Remark 21. The values of and can easily be calculated by employing the values of and , respectively, in (5).

Theorem 22. One has

Proof. Combining (23) and Lemma 3 with , we note that From (66), we deduce that Setting in Lemma 4, employing (67), and solving the resulting equation, we obtain Combining (66) and (68) we easily evaluate the values of and .

Remark 23. The values of and can easily be calculated by employing the values of and , respectively, in (5).

Theorem 24. One has where

Proof. Employing (40) in Lemma 5 with , solving the resulting equation, and noting the result from Lemma 3, we obtain where Employing (71) in Lemma 4 and solving the resulting equation, we obtain where Combining (71) and (73) we calculate the values of and .

Remark 25. The values of and can easily be calculated by employing the values of and , respectively, in (5).

Theorem 26. One has where

Proof. Employing (36) in in Lemma 5 with , solving the resulting equation, and noting the result from Lemma 3, we obtain whereEmploying (77) in Lemma 4 and solving the resulting equation, we obtain whereCombining (77) and (79) we calculate the values of and .

Remark 27. The values of and can easily be calculated by employing the values of and , respectively, in (5).

Theorem 28. One has where

Proof. Employing (37) in in Lemma 5 with , solving the resulting equation, and noting the result from Lemma 3, we obtain whereEmploying (83) in Lemma 4 and solving the resulting equation, we obtain where Combining (83) and (85) we calculate the values of and .

Remark 29. The values of and can easily be calculated by employing the values of and , respectively, in (5).

Theorem 30. One has where

Proof. Employing (38) in in Lemma 5 with , solving the resulting equation, and noting the result from Lemma 3, we obtain whereEmploying (89) in Lemma 4 and solving the resulting equation, we obtain whereCombining (89) and (91) we calculate the values of and .

Remark 31. The values of and can easily be calculated by employing the values of and , respectively, in (5).

Theorem 32. One has where

Proof. Employing (39) in Lemma 5 with , solving the resulting equation, and noting the result from Lemma 3, we obtain whereEmploying (95) in Lemma 4 and solving the resulting equation, we obtain whereCombining (95) and (97) we calculate the values of and .

Remark 33. The values of and can easily be calculated by employing the values of and , respectively, in (5).

Remark 34. (i) From (36) and (77) we can easily deduce the values of class invariants and . (ii) Combining (37) and (83) we can calculate the values of class invariants and . (iii) The values of class invariants and can be calculated from (38) and (89). (iv) Combining (39) and (95) we can obtain the values of class invariants and . (v) Combining (40), (71), and Lemma 3 we can calculate the values of class invariants and .

Acknowledgment

The author is thankful to University Grants Commission, New Delhi, India, for partially supporting the research work under the Grant no. F. No. 41-1394/2012(SR).

References

  1. S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, India, 1988.
  2. 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. View at: Publisher Site | Google Scholar
  3. G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook. Part II, Springer, New York, NY, USA, 2009.
  4. K. G. Ramanathan, “On some theorems stated by Ramanujan,” in Number Theory and Related Topics (Bombay, 1988), vol. 12, pp. 151–160, Oxford University Press, Bombay, India, 1989. View at: Google Scholar
  5. J. Yi, Construction and application of modular equation [Ph.D. thesis], University of Illionis, Chicago, Ill, USA, 2001.
  6. J. Yi, “Some new modular equations and their applications,” Journal of Mathematical Analysis and Applications, vol. 319, no. 2, pp. 531–546, 2006. View at: Publisher Site | Google Scholar
  7. N. D. Baruah and N. Saikia, “Some general theorems on the explicit evaluations of Ramanujan's cubic continued fraction,” The Journal of Computational and Applied Mathematics, vol. 160, no. 1-2, pp. 37–51, 2003. View at: Google Scholar
  8. N. D. Baruah and N. Saikia, “Some new explicit values of Ramanujan's continued fractions,” Indian Journal of Mathematics, vol. 46, no. 2-3, pp. 197–222, 2004. View at: Google Scholar
  9. M. S. M. Naika, M. C. Maheshkumar, and K. S. Bairy, “Certain quotient of eta-function identities,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 1, pp. 121–136, 2008. View at: Google Scholar
  10. B. C. Berndt, Ramanujan's Notebooks. V, Springer, New York, NY, USA, 1998. View at: Publisher Site
  11. N. D. Baruah, “On some class invariants of Ramanujan,” The Journal of the Indian Mathematical Society. New Series, vol. 68, no. 1–4, pp. 113–131, 2001. View at: Google Scholar
  12. N. Saikia, “Ramanujan's Schläfli-type modular equations and class invariants gn,” Functiones et Approximatio Commentarii Mathematici. In press. View at: Google Scholar
  13. N. Saikia, “Ramanujan's modular equations and Weber-Ramanujan class invariants Gn and gn,” Bulletin of Mathematical Sciences, vol. 2, no. 1, pp. 205–223, 2012. View at: Publisher Site | Google Scholar
  14. N. Saikia, “A parameter for Ramanujans function χ(q): its explicit values and applications,” ISRN Computational Mathematics, vol. 2012, Article ID 169050, 14 pages, 2012. View at: Publisher Site | Google Scholar
  15. B. C. Berndt, Ramanujan's Notebooks. III, Springer, New York, NY, USA, 1991. View at: Publisher Site

Copyright © 2013 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.

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