Rogers–Ramanujan identities

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In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series, first discovered and proved by Leonard James Rogers (1894). They were subsequently rediscovered (without a proof) by Srinivasa Ramanujan some time before 1913. Ramanujan had no proof, but rediscovered Rogers's paper in 1917, and they then published a joint new proof (Rogers & Ramanujan 1919). Issai Schur (1917) independently rediscovered and proved the identities.


The Rogers–Ramanujan identities are

G(q) = \sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} = 
\frac {1}{(q;q^5)_\infty (q^4; q^5)_\infty}
	=1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots \,
(sequence A003114 in OEIS)


H(q) =\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} = 
\frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty}
=1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots \,
(sequence A003106 in OEIS).

Here, (\cdot;\cdot)_n denotes the q-Pochhammer symbol.

Modular functions[edit]

If q = e2πiτ, then q−1/60G(q) and q11/60H(q) are modular functions of τ.


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

Ramanujan's continued fraction is

1+\frac{q}{1+\frac{q^2}{1+\frac{q^3}{1+\cdots}}}  = \frac{G(q)}{H(q)}.

See also[edit]


  • Rogers, L. J.; Ramanujan, Srinivasa (1919), "Proof of certain identities in combinatory analysis.", Cambr. Phil. Soc. Proc. 19: 211–216, Reprinted as Paper 26 in Ramanujan's collected papers 
  • Rogers, L. J. (1892), "On the expansion of some infinite products", Proc. London Math. Soc. 24 (1): 337–352, doi:10.1112/plms/s1-24.1.337, JFM 25.0432.01 
  • Rogers, L. J. (1893), "Second Memoir on the Expansion of certain Infinite Products", Proc. London Math. Soc. 25 (1): 318–343, doi:10.1112/plms/s1-25.1.318 
  • Rogers, L. J. (1894), "Third Memoir on the Expansion of certain Infinite Products", Proc. London Math. Soc. 26 (1): 15–32, doi:10.1112/plms/s1-26.1.15 
  • 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 identities of the Rogers-Ramanujan type", Proceedings of the London Mathematical Society. Second Series 54 (2): 147–167, doi:10.1112/plms/s2-54.2.147, ISSN 0024-6115, MR 0049225 

External links[edit]