Turkish Journal of Analysis and Number Theory
ISSN (Print): 2333-1100 ISSN (Online): 2333-1232 Website: http://www.sciepub.com/journal/tjant
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Turkish Journal of Analysis and Number Theory. 2015, 3(2), 37-42
DOI: 10.12691/tjant-3-2-1
Open AccessArticle

The Rogers-Ramanujan Identities

Fazlee Hossain1, Sabuj Das2 and Haradhan Kumar Mohajan3,

1Department of Mathematics, University of Chittagong, Bangladesh

2Department of Mathematics, Raozan University College, Bangladesh

3Premier University, Chittagong, Bangladesh

Pub. Date: April 01, 2015

Cite this paper:
Fazlee Hossain, Sabuj Das and Haradhan Kumar Mohajan. The Rogers-Ramanujan Identities. Turkish Journal of Analysis and Number Theory. 2015; 3(2):37-42. doi: 10.12691/tjant-3-2-1

Abstract

In 1894, Rogers found the two identities for the first time. In 1913, Ramanujan found the two identities later and then the two identities are known as The Rogers-Ramanujan Identities. In 1982, Baxter used the two identities in solving the Hard Hexagon Model in Statistical Mechanics. In 1829 Jacobi proved his triple product identity; it is used in proving The Rogers-Ramanujan Identities. In 1921, Ramanujan used Jacobi’s triple product identity in proving his famous partition congruences. This paper shows how to generate the generating function for C'(n), C1'(n), C''(n), and C1''(n), and shows how to prove the Corollaries 1 and 2 with the help of Jacobi’s triple product identity. This paper shows how to prove the Remark 3 with the help of various auxiliary functions and shows how to prove The Rogers-Ramanujan Identities with help of Ramanujan’s device of the introduction of a second parameter a.

Keywords:
at most auxiliary function convenient expansion minimal difference operator Ramanujan’s device

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References:

[1]  Andrews, G.E, “An Introduction to Ramanujan’s Lost Notebook”, American Mathmatical Monthly, 86: 89-108. 1979.
 
[2]  Hardy, G.H. and Wright, E.M. “Introduction to the Theory of Numbers”, 4th Edition, Oxford, Clarendon Press, 1965.
 
[3]  Jacobi, C.G.J. (1829), “Fundamenta Nova Theoriae Functionum Ellipticarum (in Latin), Konigsberg Borntraeger, Cambridge University Press, 2012.
 
[4]  Baxter, R.J., “Exactly Solved Model in Statistical Models”, London, Academic Press, 1982.
 
[5]  Ramanujan, S., “Congruence Properties of Partitions”, Math, Z. 9: 147-153. 1921.
 
[6]  Ramanujan, S., “Some Properties of P(n), Number of Partitions of n”, Proc. of the Cam. Philo. Society XIX, 207-210. 1919.
 
[7]  Das, S. and Mohajan, H.K., “Generating Function for P(n,p,*) and P(n, *,p)”, Amer. Rev. of Math. and Sta. 2(1): 33-35. 2014.