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.

A Framework for the Identities[edit]

In April 2014 Ken Ono, a number theorist at Emory University, announced that he had found a framework for the Rogers-Ramanujan identities and their arithmetic properties, solving a long-standing mystery stemming from the work of Ramanujan. The findings yield a treasure trove of algebraic numbers and formulas to access them. Ono has two co-authors for this work, S. Ole Warnaar of the University of Queensland and Michael Griffin, an Emory University graduate student.


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]