Chandrashekar Adiga and A. Vanitha

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 20, 2014, Number 1, Pages 36—48

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## Details

### Authors and affiliations

Chandrashekar Adiga

*Department of Studies in Mathematics
University of Mysore
Manasagangotri, Mysore 570 006, India
*

A. Vanitha

*Department of Studies in Mathematics
University of Mysore
Manasagangotri, Mysore 570 006, India
*

### Abstract

In this paper, we establish two modular relations for the Rogers–Ramanujan–Slater functions of order fifteen. These relations are analogues to Ramanujan’s famous forty identities for the Rogers–Ramanujan functions.

Furthermore, we give interesting partition theoretic interpretations of these relations.

### Keywords

- Rogers–Ramanujan functions
- Theta functions
- Jacobi’s triple product identity
- Colored partitions

### AMS Classification

- 33D15
- 11P83

### References

- Adiga, C., N. A. S. Bulkhali, On certain new modular relations for the Rogers–Ramanujan type functions of order ten and its apllications to partitions (Submitted for publication).
- Adiga, C., N. A. S. Bulkhali, Some modular relations analogues to the Ramanujan’s forty identities with its applications to partitions, Axioms, Vol. 2, 2013, 20–43.
- Adiga, C., K. R. Vasuki, N. Bhaskar, Some new modular relations for the cubic functions, South East Asian Bull. Math., Vol. 36, 2012, 1–19.
- C. Adiga, K. R. Vasuki, B. R. Srivatsa Kumar, On modular relations for the functions analogous to Rogers–Ramanujan functions with applications to partitions, South East J. Math. and Math. Sc., Vol. 6, 2008, NO. 2, 131–144.
- C. Adiga, B. C. Berndt, S. Bhargava, G. N. Watson, Chapter 16 of Ramanujan’s second notebook: Theta functions and q-series, Mem. Amer. Math. Soc., Vol. 315, 1985, 1–91.
- Baruah, N. D., J. Bora, Further analogues of the Rogers–Ramanujan functions with applications to partitions, Elec. J. combin. Number Thy., Vol. 7, 2007, No. 2, 1–22.
- Baruah, N. D., J. Bora, Modular relations for the nonic analogues of the Rogers–Ramanujan functions with applications to partitions, J. Number Thy., Vol. 128, 2008, 175–206.
- Baruah, N. D., J. Bora, N. Saikia, Some new proofs of modular relations for the Göllnitz–Gordon functions, Ramanujan J.,Vol. 15, 2008, 281–301.
- Berndt, B. C., G. Choi, Y. S. Choi, H. Hahn, B. P. Yeap, A. J. Yee, H. Yesilyurt, J. Yi, Ramanujan’s forty identities for the Rogers–Ramanujan function, Mem., Amer. Math. Soc., Vol. 188 (880), 2007, 1–96.
- Berndt, B. C., H. Yesilyurt, New identities for the Rogers–Ramanujan function, Acta Arith., Vol. 120, 2005, 395–413.
- Biagioli, A. J. F. A proof of some identities of Ramanujan using modular functions, Glasg. Math. J., Vol. 31, 1989, 271–295.
- Birch, B. J. A look back at Ramanujan’s Notebooks, Math. Proc. Camb. Soc., Vol. 78, 1975, 73–79.
- Bressoud, D. Proof and generalization of certain identities conjectured by Ramanujan, Ph.D. Thesis, Temple University, 1977.
- Chen, S.-L., S.-S. Huang, New modular relations for the Göllnitz-Gordon functions, J. Number Thy., Vol. 93, 2002, 58–75.
- Gugg, C. Modular identities for the Rogers–Ramanujan functions and Analogues, Ph.D. Thesis, University of Illinois at Urbana-Champaign, 2010.
- Hahn, H. Septic analogues of the Rogers–Ramanujan functions, Acta Arith., Vol. 110, 2003, No. 4, 381–399.
- Hahn, H. Eisenstein Series, Analogues of the Rogers–Ramanujan functions, and Partition Identities, Ph.D. Thesis, University of Illinois at Urbana-Champaign, 2004.
- Huang, S.-S. On modular relations for the Göllnitz-Gordon functions with applications to partitions, J. Number Thy., Vol. 68, 1998, 178–216.
- Mc Laughlin, J., A. V. Sills, P. Zimmer, Rogers–Ramanujan–Slater type identities, Elec. J. Combin., Vol. 15, 2008, 1–59.
- Ramanujan, S. Algebraic relations between certain infinite products, Proc. London Math. Soc., Vol. 2, 1920, p. xviii.
- Ramanujan, S. The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.
- Rogers, L. J. On two theorems of combinatory analysis and some allied identities, Proc. London Math. Soc., Vol. 16, 1917, No. 2, 315–336.
- Rogers, L. J. On a type of modular relation, Proc. London Math. Soc., Vol. 19, 1921, 387– 397.
- Slater, L. J. Further identities of Rogers–Ramanujan type, London Math. Soc., Vol. 54, 1952, No. 2, 147–167.
- Vasuki, K. R., P. S. Guruprasad, On certain new modular relations for the Rogers– Ramanujan type functions of order twelve, Adv. Stud. Contem. Math., Vol. 20, 2000, No. 3, 319–333.
- Vasuki, K. R., G. Sharath, K. R. Rajanna, Two modular equations for squares of the cubicfunctions with applcations, Note di Matematica, Vol. 30, 2010, No. 2, 61–71.
- Watson, G. N. Proof of certain identities in combinatory analysis, J. Indian Math. Soc., Vol. 20, 1933, 57–69.

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## Cite this paper

APAAdiga, C. & Vanitha, A. (2014). New modular relations for the Rogers–Ramanujan type functions of order fifteen. Notes on Number Theory and Discrete Mathematics, 20(1), 36-48.

ChicagoAdiga, Chandrashekar, and A. Vanitha. “New Modular Relations for the Rogers–Ramanujan Type Functions of Order Fifteen.” Notes on Number Theory and Discrete Mathematics 20, no. 1 (2014): 36-48.

MLAAdiga, Chandrashekar, and A. Vanitha. “New Modular Relations for the Rogers–Ramanujan Type Functions of Order Fifteen.” Notes on Number Theory and Discrete Mathematics 20.1 (2014): 36-48. Print.