The Rogers-Ramanujan Identities

Hossain, Fazlee and Das, Sabuj (2015): The Rogers-Ramanujan Identities. Published in: Turkish Journal of Analysis and Number Theory , Vol. 3, No. 2 (1 April 2015): pp. 37-42.

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 , , and , 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.

• Item Type: MPRA Paper
• Language: English
• Keywords: At most, auxiliary function, convenient, expansion, minimal difference, operator, Ramanujan’s device.
• Subjects: C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables; C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C30 - General

References

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[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.

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