Rogers-Ramanujan Identities

For and using the Notation of the Ramanujan Theta Function, the Rogers-Ramanujan identities are

 (1)

 (2)

where are q-Series. Written out explicitly (Hardy 1959, p. 13),

 (3)

 (4)

The identities can also be written succinctly as

 (5)

where , 1.

Other forms of the Rogers-Ramanujan identities include

 (6)

and
 (7)

(Petkovsek et al. 1996).

References

Andrews, G. E. The Theory of Partitions. Cambridge, England: Cambridge University Press, 1985.

Andrews, G. E. -Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., pp. 17-20, 1986.

Andrews, G. E. and Baxter, R. J. A Motivated Proof of the Rogers-Ramanujan Identities.'' Amer. Math. Monthly 96, 401-409, 1989.

Bressoud, D. M. Analytic and Combinatorial Generalizations of the Rogers-Ramanujan Identities. Providence, RI: Amer. Math. Soc., 1980.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 13, 1959.

Paule, P. Short and Easy Computer Proofs of the Rogers-Ramanujan Identities and of Identities of Similar Type.'' Electronic J. Combinatorics 1, R10 1-9, 1994. http://www.combinatorics.org/Volume_1/volume1.html#R10.

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, p. 117, 1996.

Robinson, R. M. Comment to: A Motivated Proof of the Rogers-Ramanujan Identities.''' Amer. Math. Monthly 97, 214-215, 1990.

Rogers, L. J. Second Memoir on the Expansion of Certain Infinite Products.'' Proc. London Math. Soc. 25, 318-343, 1894.

Sloane, N. J. A. Sequence A006141/M0260 in `An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.