Rogers-Ramanujan identities

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

[edit] Definition

The Rogers-Ramanujan identities are

$\sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} = \frac {1}{(q;q^5)_\infty (q^4; q^5)_\infty}$

and

$\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} = \frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty}.$

Here, $(a;q)_\infty$ denotes the infinite q-Pochhammer symbol.

Leonard James Rogers, Second memoir on the expansion of certain infinite products (1894) Proceedings of the London Mathematical Society, 25, pp 318-343.
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.