New approach for Somos’s Dedekind eta-function identities of level 6
DOI:
https://doi.org/10.22199/issn.0717-6279-2021-01-0012Keywords:
Modular equation, Dedekind eta-function, Colored partitionsAbstract
In the present work, we prove few new Dedekind eta-function identities of level 6 discovered by Somos in two different methods. Also during this process, we give an alternate method to Somos’s Dedekind eta-function identities of level 6 proved by B. R. Srivatsa Kumar et al. As an application of this, we establish colored partition identities.
References
C. Adiga, M. S. M. Naika, and K. Shivashankara, “On some P-Q eta-function identities of Ramanujan”, Indian journal of mathematics, vol. 44, no. 3, pp. 253-267, Dec. 2002.
C. Adiga, T. Kim, M. S. M. Naika, and H. S. Madhusudhan, “On Ramanujan’s cubic continued fraction and explicit evaluations of theta-functions”, Indian journal pure applications mathematics, vol. 35, no. 9, pp. 1047-1062, Sep. 2004. [On line]. Available: https://bit.ly/39JxcpB
C. Adiga, N. A. S Bulkhali, D. Ranganatha, and H. M. Srivastava, “Some new modular relations for the Rogers-Ramanujan type functions of order eleven with applications to partitions”, Journal of number theory, vol. 158, pp. 281-297, Jan. 2016, doi: 10.1016/j.jnt.2015.06.019
N. D. Baruah, “Modular equations for Ramanujan’s cubic continued fraction”, Journal mathematics analysis applications, vol. 268, no. 1, pp. 244-255, Apr. 2002, doi: 10.1006/jmaa.2001.7823
N. D. Baruah, “On some of Ramanujan’s Schläfli-type ‘mixed’ modular equations”, Journal number theory, vol. 100, no. 2, pp. 270-294, Jun. 2003, doi: 10.1016/S0022-314X(02)00127-0
B. C. Berndt, Ramanujan's notebooks, vol. 3. New York, NY: Springer, 1991, doi: 10.1007/978-1-4612-0965-2
B. C. Berndt, Ramanujan's notebooks, vol. 4. New York, NY: Springer, 1996, doi: 10.1007/978-1-4612-0879-2
Z. Cao, “On Somos’s dissection identities”, Journal of mathematical analysis and applications, vol. 365, no. 2, pp. 659-667, May 2010, doi: 10.1016/j.jmaa.2009.11.038
S. -S. Huang, “On modular relation for the Göllnitz-Gordon functions with applications to partitions”, Journal number theory, vol 68, no. 2, pp. 178-216, Feb. 1998, doi: 10.1006/jnth.1997.2205
M. S. M. Naika, “A note on cubic modular equations of degree two *”, Tamsui Oxford Journal Mathematics Society, vol. 22, no. 1, pp. 1-8, 2006. [On line]. Available: https://bit.ly/2LyiTfb
S. Ramanujan, Notebooks, 2 vols. Bombay: Tata Institute of Fundamental Research, 1957.
S. Ramanujan, The lost notebook and other unpublished papers, New Delhi: Narosa, 1988.
N. Saikia, “Modular identities and explicit values of a continued fraction of order twelve”, JP journal of algebra, number theory and applications, vol. 22, no 2, pp. 127-154, Sep. 2011. [On line]. Available: https://bit.ly/3nEGEPW
M. Somos, Personal communication [e-mail], October 24, 2004.
B. R. Srivatsa Kumar and R. G. Veeresha, “Partition identities arising from Somos’s theta function identities”, Annali Dell ‘Universita’ Di Ferrara, vol. 63, pp. 303-313, Nov. 2017, doi: 10.1007/s11565-016-0261-z
B. R. Srivatsa Kumar and D. Anu Radha, “Somos’s theta-function identities of level 10”, Turkish journal of mathematics, vol. 42, pp. 763-773, May 2018, doi: 10.3906/mat-1703-51
B. R. Srivatsa Kumar, K. R. Rajanna, and R. Narendra, “New theta-function identities of level 6 in the spirit of Ramanujan”, Mathematical notes, vol. 106, no. 6, pp. 922-929, Nov. 2019, doi: 10.1134/S0001434619110282
B. R. Srivatsa Kumar, K. R. Rajanna, and R. Narendra, “Theta-function identities of level 8 and its application to partitions”, Afrika matematika, vol. 30, no. 1-2, pp. 257-267, Mar. 2019, doi: 10.1007/s13370-018-0643-8
K. R. Vasuki, T. G. Sreeramamurthy, “A note on P-Q modular equations”, Tamsui Oxford Journal Mathematics Sciences, vol. 21, no. 2, pp. 109-120, 2005. [On line]. Available: https://bit.ly/3iaeaMI
K. R. Vasuki, “On some of Ramanujan’s P-Q modular equations”, Journal Indian Mathematical Society, vol. 73, no. 3-4, pp. 131-143, 2006.
K. R. Vasuki and R. G. Veeresha, “On Somos’s theta-function identities of level 14”, Ramanujan journal, vol. 42, pp. 131-144, Jan. 2017, doi: 10.1007/s11139-015-9714-8
J. Yi, “Theta-function identities and the explicit formulas for theta function and their applications”, Journal mathematics analysis applications, vol. 292, no. 2, pp. 381-400, Apr. 2004, doi: 10.1016/j.jmaa.2003.12.009
B. Yuttanan, “New modular equations in the spirit of Ramanujan”, Ramanujan journal, vol. 29, pp. 257-272, Dec. 2012, doi: 10.1007/s11139-012-9399-1
Published
How to Cite
Issue
Section
Copyright (c) 2021 D. Anu Radha, B. R. Srivatsa Kumar, Shruthi
This work is licensed under a Creative Commons Attribution 4.0 International License.
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
