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| In [[mathematics]], the '''Rogers–Ramanujan identities''' are two identities related to [[basic hypergeometric series]], first discovered and proved by {{harvs|txt|first=Leonard James|last= Rogers|authorlink=Leonard James Rogers|year=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 {{harv|Rogers|Ramanujan|1919}}. {{harvs|txt|first=Issai |last=Schur|authorlink=Issai Schur|year=1917}} independently rediscovered and proved the identities.
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| ==Definition==
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| The Rogers–Ramanujan identities are
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| :<math>G(q) = \sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} =
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| \frac {1}{(q;q^5)_\infty (q^4; q^5)_\infty}
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| =1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots \,
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| </math> {{OEIS|A003114}}
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| and
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| :<math>H(q) =\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} =
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| \frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty}
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| =1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots \,
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| </math> {{OEIS|A003106}}.
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| Here, <math>(\cdot;\cdot)_n</math> denotes the [[q-Pochhammer symbol]].
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| ==Modular functions==
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| If ''q'' = e<sup>2πiτ</sup>, then ''q''<sup>−1/60</sup>''G''(''q'') and ''q''<sup>11/60</sup>''H''(''q'') are [[modular function]]s of τ.
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| ==Applications==
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| The Rogers–Ramanujan identities appeared in Baxter's solution of the [[hard hexagon model]] in statistical mechanics.
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| [[Ramanujan's continued fraction]] is
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| :<math>1+\frac{q}{1+\frac{q^2}{1+\frac{q^3}{1+\cdots}}} = \frac{G(q)}{H(q)}.</math>
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| ==See also==
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| *[[Rogers polynomials]]
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| ==References==
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| *{{Citation | last1=Rogers | first1=L. J. | last2=Ramanujan | first2=Srinivasa | author2-link=Srinivasa Ramanujan | title=Proof of certain identities in combinatory analysis. | id=Reprinted as Paper 26 in Ramanujan's collected papers | year=1919 | journal=Cambr. Phil. Soc. Proc. | volume=19 | pages=211–216}}
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| *{{Citation | last1=Rogers | first1=L. J. | title=On the expansion of some infinite products | doi=10.1112/plms/s1-24.1.337 | jfm=25.0432.01 | year=1892 | journal=Proc. London Math. Soc. | volume=24 | issue=1 | pages=337–352 }}
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| *{{Citation | last1=Rogers | first1=L. J. | title=Second Memoir on the Expansion of certain Infinite Products | doi=10.1112/plms/s1-25.1.318 | year=1893 | journal=Proc. London Math. Soc. | volume=25 | issue=1 | pages=318–343}}
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| *{{Citation | last1=Rogers | first1=L. J. | title=Third Memoir on the Expansion of certain Infinite Products | doi=10.1112/plms/s1-26.1.15 | year=1894 | journal=Proc. London Math. Soc. | volume=26 | issue=1 | pages=15–32}}
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| * Issai Schur, ''Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche'', (1917) Sitzungsberichte der Berliner Akademie, pp. 302–321.
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| * [[W.N. Bailey]], ''Generalized Hypergeometric Series'', (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.
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| * 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.
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| * [[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.
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| * 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.
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| *{{Citation | last1=Slater | first1=L. J. | title=Further identities of the Rogers-Ramanujan type | doi=10.1112/plms/s2-54.2.147 | mr=0049225 | year=1952 | journal=Proceedings of the London Mathematical Society. Second Series | issn=0024-6115 | volume=54 | issue=2 | pages=147–167}}
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| ==External links==
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| *{{mathworld|urlname=Rogers-RamanujanIdentities|title=Rogers-Ramanujan Identities}}
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| *{{mathworld|urlname=Rogers-RamanujanContinuedFraction|title=Rogers-Ramanujan Continued Fraction}}
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| {{DEFAULTSORT:Rogers-Ramanujan identities}}
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| [[Category:Hypergeometric functions]]
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| [[Category:Mathematical identities]]
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| [[Category:Q-analogs]]
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| [[Category:Modular forms]]
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| [[Category:Srinivasa Ramanujan]]
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| {{numtheory-stub}}
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