Rogers–Ramanujan identities
- Rogers–Ramanujan identities
In mathematics, the Rogers–Ramanujan identities are a set of identities related to basic hypergeometric series. They were discovered by harvs|txt|first=Leonard James|last= Rogers|authorlink=Leonard James Rogers|year=1894 and subsequently rediscovered by harvs|txt|first=Srinivasa|last= Ramanujan|authorlink=Srinivasa Ramanujan|year=1913 as well as by harvs|txt|first=Issai |last=Schur|authorlink=Issai Schur|year=1917.
Definition
The Rogers–Ramanujan identities are
:
and
:
Here, denotes the infinite q-Pochhammer symbol.
References
*
* 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, " [http://www.math.uiuc.edu/~berndt/articles/rrcf.pdf The Rogers-Ramanujan Continued Fraction] ", J. Comput. Appl. Math. 105 (1999), pp. 9–24.
* Cilanne Boulet, Igor Pak, " [http://www-math.mit.edu/~pak/rogers14.pdf A Combinatorial Proof of the Rogers-Ramanujan and Schur Identities] ", Journal of Combinatorial Theory, Ser. A, vol. 113 (2006), 1019–1030.
*Citation | last1=Slater | first1=L. J. | title=Further identies of the Rogers-Ramanujan type | doi=10.1112/plms/s2-54.2.147 | id=MathSciNet | id = 0049225 | year=1952 | journal=Proceedings of the London Mathematical Society. Second Series | issn=0024-6115 | volume=54 | pages=147–167
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