Rogers–Ramanujan identities

In mathematics, the Rogers–Ramanujan identities are a set of identities related to basic hypergeometric series. They were discovered by Leonard James Rogers (1894) and subsequently rediscovered by Srinivasa Ramanujan (1913) as well as by Issai Schur (1917).

Definition

The Rogers–Ramanujan identities are

(sequence A003114 in OEIS)

and

(sequence A003106 in OEIS)

Here, denotes the infinite q-Pochhammer symbol.

Modular functions

If q = e^2πiτ, then q^−1/60G(q) and q^11/60H(q) are modular functions of τ.

Applications

The Rogers-Ramanujan identites appeared in Baxter's solution of the hard hexagon model in statistical mechanics.

Ramanujan's continued fraction is

References

• Rogers, L. J. (1894), "Second Memoir on the Expansion of certain Infinite Products", Proc. London Math. Soc. s1-25: 318–343, doi:10.1112/plms/s1-25.1.318 
• Issai Schur, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche, (1917) Sitzungsberichte der Berliner Akademie, pp. 302–321.
• W.N. Bailey, Generalized Hypergeometric Series, (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.
• George Gasper and Mizan Rahman, Basic Hypergeometric Series, 2nd Edition, (2004), Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. ISBN 0-521-83357-4.
• Bruce C. Berndt, Heng Huat Chan,, Sen-Shan Huang, Soon-Yi Kang, Jaebum Sohn, Seung Hwan Son, The Rogers-Ramanujan Continued Fraction, J. Comput. Appl. Math. 105 (1999), pp. 9–24.
• Cilanne Boulet, Igor Pak, A Combinatorial Proof of the Rogers-Ramanujan and Schur Identities, Journal of Combinatorial Theory, Ser. A, vol. 113 (2006), 1019–1030.
• Slater, L. J. (1952), "Further identies of the Rogers-Ramanujan type", Proceedings of the London Mathematical Society. Second Series 54: 147–167, doi:10.1112/plms/s2-54.2.147, MR0049225, ISSN 0024-6115 

External links

Weisstein, Eric W., "Rogers-Ramanujan Identities" from MathWorld. Weisstein, Eric W., "Rogers-Ramanujan Continued Fraction" from MathWorld.