Rogers–Ramanujan identities

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.

Definition

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 the OEIS)

and

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

Here, $(\cdot ;\cdot )_{n}$ denotes the q-Pochhammer symbol.

Integer Partitions

Consider the following:

• ${\displaystyle {\frac {q^{n^{2}}}{(q;q)_{n}}}}$ is the generating function for partitions with exactly $n$ parts such that adjacent parts have difference at least 2.
• ${\displaystyle {\frac {1}{(q;q^{5})_{\infty }(q^{4};q^{5})_{\infty }}}}$ is the generating function for partitions such that each part is congruent to either 1 or 4 modulo 5.
• ${\displaystyle {\frac {q^{n^{2}+n}}{(q;q)_{n}}}}$ is the generating function for partitions with exactly $n$ parts such that adjacent parts have difference at least 2 and such that the smallest part is at least 2.
• ${\displaystyle {\frac {1}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}}$ is the generating function for partitions such that each part is congruent to either 2 or 3 modulo 5.

The Rogers–Ramanujan identities could be now interpreted in the following way. Let $n$ be a non-negative integer.

1. The number of partitions of $n$ such that the adjacent parts differ by at least 2 is the same as the number of partitions of $n$ such that each part is congruent to either 1 or 4 modulo 5.
2. The number of partitions of $n$ such that the adjacent parts differ by at least 2 and such that the smallest part is at least 2 is the same as the number of partitions of $n$ such that each part is congruent to either 2 or 3 modulo 5.

Alternatively,

1. The number of partitions of $n$ such that with $k$ parts the smallest part is at least $k$ is the same as the number of partitions of $n$ such that each part is congruent to either 1 or 4 modulo 5.
2. The number of partitions of $n$ such that with $k$ parts the smallest part is at least $k+1$ is the same as the number of partitions of $n$ such that each part is congruent to either 2 or 3 modulo 5.

Modular functions

If q = e^2πiτ, then q^−1/60G(q) and q^11/60H(q) are modular functions of τ.

Applications

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

• Rogers polynomials

References

• 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
• Schur, Issai (1917), "Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche", Sitzungsberichte der Berliner Akademie: 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

• Weisstein, Eric W. "Rogers-Ramanujan Identities". MathWorld.

• Weisstein, Eric W. "Rogers-Ramanujan Continued Fraction". MathWorld.