| 释义 |
Rogers-Ramanujan IdentitiesFor  and using the Notation of the Ramanujan Theta Function,the Rogers-Ramanujan identities are
 | (1) |
 | (2) |
 are q-Series. Written out explicitly (Hardy 1959, p. 13),
 | (3) |
 | (4) |
 | (5) |
 , 1.
Other forms of the Rogers-Ramanujan identities include
 | (6) |
 | (7) |
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. SequenceA006141/M0260in ``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. |