The Rogers-Ramanujan identities were initially stated and proved by L.J. Rogers [R1] as a pair of analytic basic hypergeometric identities. These state that if $|q| < 1$, then \begin{equation}\label{rreq1} \sum_{n=0}^\infty \frac{q^{n^2}}{(q;q)_n} = \frac{1}{(q,q^4;q^5)_{\infty}}; \end{equation} \begin{equation}\label{rreq2} \sum_{n=0}^\infty \frac{q^{n(n+1)}}{(q;q)_n} = \frac{1}{(q^2,q^3;q^5)_{\infty}}. \end{equation} They were rediscovered by Ramanujan sometime before 1913, and were rediscovered and proved independently by Issai Schur[Sc1]. There are now many different proofs. In particular, Lucy Slater proved them using the method of Bailey pairs, in her compendium of 130 such identities published in [Sl1] and [Sl2].

When the right and left sides of each identity at \eqref{rreq1} and \eqref{rreq2} are regarded as the generating functions for certain types of restricted partitions, the following partition identities follow respectively from the above analytic identities ($n$ is any positive integer):

1) The number of partitions of $n$ into distinct parts with gap at least 2 between consecutive parts is the same as the number of partitions of $n$ into parts congruent to either 1 or 4 modulo 5.

2) The number of partitions of $n$ into distinct parts with gap at least 2 between consecutive parts and no part equal to 1 is the same as the number of partitions of $n$ into parts congruent to either 2 or 3 modulo 5.

Alternatively, by considering the conjugate of the first type of partition in each case, the following alternative version of these identities follow:

1') The number of partitions of $n$ in which every part from 1 to the largest part occurs at least least twice is the same as the number of partitions of $n$ into parts congruent to either 1 or 4 modulo 5.

2') The number of partitions of $n$ in which every part smaller than the largest part occurs at least least twice is the same as the number of partitions of $n$ into parts congruent to either 2 or 3 modulo 5.

Interestingly, there is still no simple bijective proof of the Rogers-Ramanujan identities.

[R1] L.J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), pp. 318-343.

[Sc1] I. Schur, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüchen, in Gesammelte Abhandlungen. Band II, Springer-Verlag, Berlin-New York, 1973, 117-136.
(Originally in Sitzungsberichte der Preussischen Akadamie derWissenschaften, 1917, Physikalisch-Mathematische Klasse, 302-321)

[Sl1] L. J. Slater, A new proof of Rogers's transformations of infinite series, Proc. London Math. Soc. (2) 53 (1951), 460-475.

[Sl2] L. J. Slater, Further identities of the Rogers-Ramanujan type, Proc. London Math.Soc. (2) 54 (1952), 147-167.

Back to the main Partitions page