(Petkovšek et al. 1996). Even though they appear to be infinite series, only finitely many of the terms on each side are nonzero since when . (5) and (6)
specify a sequence of polynomials, indexed by , as gets larger and larger, the polynomial approximates the series
(◇) more and more closely. Also, taking the limit as recovers (◇) after applying the Jacobi
triple product identity on the right hand side. Finally, (5)
and (6) are actually equivalent to each other; (6)
is the "symmetrized" version of (5) using the so-called
Paule symmetrization.
These identities can also be written even more succinctly as the single identity
(7)
for ,
1.
The formulas have a curious history, having been proved by Rogers (1894) in a paper that was completely ignored, then rediscovered (without proof) by Ramanujan sometime
before 1913. The formulas were communicated to MacMahon, who published them in his
famous text, still without proof. Then, in 1917, Ramanujan accidentally found Roger's
1894 paper while leafing through a journal. In the meantime, Schur (1917) independently
rediscovered and published proofs for the identities (Hardy 1999, p. 91). Garsia
and Milne (1981ab) gave the first proof of the Rogers-Ramanujan identities to construct
a bijection between the relevant classes of partitions
(Andrews 1986, p. 59).
Bailey (1947, 1949) systematically studied and generalized Rogers's work on Rogers-Ramanujan type identities.
Slater (1952) published a list of 130 identities of Rogers-Ramanujan type, some of which were already known, but many which were new and due to Slater. A few of these are summarized in the following table. Note that Slater's tabulation actually contained a number of identities listed twice, and a few listed three times, as a result of two different starting points sometimes leading to the same final result, but with a possibly with a slightly different algebraic presentation.
Schur showed that (◇) has the combinatorial interpretation that the number of partitions of with minimal difference is equal to the number of partitions into parts of
the forms or (Hardy 1999, p. 92). The following table gives the
first few values.
min. diff.
(mod 5)
1
1
1
1
2
1
2
3
1
3
4
2
4,
4,
5
2
5,
,
6
3
6, ,
6, ,
There is a similar combinatorial interpretation for (◇).
4. The five identities due to Andrews (1975) of type (triple product on mod 11 over ),
but the series representations are double series and therefore not as elegant as
the other identities.
5. The six double series expansions of type mod 13 over type products.
Here, "sort of" refers to the fact that between and , there is an "identity" in which the product
side contains , so the identity reduces
to
and therefore is not listed.
There is also a different sequence of identities given by
1. The Rogers-Ramanujan identities (2 identities mod ).
The next in the sequence would be 5 identities with modulus . A. Sills worked out a series expansion these
identities, but it was so ugly that he did not publish it (A. Sills, pers. comm.,
Mar. 16, 2005).
Andrews, G. E. "On Rogers-Ramanujan Type Identities Related to the Moduls 11." Proc. London Math. Soc.30, 330-346,
1975.Andrews, G. E. "The Hard-Hexagon Model and Rogers-Ramanujan
Type Identities." Proc. Nat. Acad. Sci. U.S.A.78, 5290-5292,
1981.Andrews, G. E. Encyclopedia
of Mathematics and Its Applications, Vol. 2: The Theory of Partitions.
Cambridge, England: Cambridge University Press, pp. 109 and 238, 1984.Andrews,
G. E. q-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. Monthly96, 401-409, 1989.Andrews,
G. E. and Santos, J. P. O. "Rogers-Ramanujan Type Identities
for Partitions with Attached Odd Parts." Ramanujan J.1, 91-99,
1997.Andrews, G. E.; Baxter, R. J.; and Forrester, P. J.
"Eight-Vertex SOS Model and Generalized Rogers-Ramanujan-Type Identities."
J. Stat. Phys.35, 193-266, 1984.Bailey, W. N. "Some
Identities in Combinatory Analysis." Proc. London Math. Soc.49,
421-435, 1947.Bailey, W. N, "Identities of the Rogers-Ramanujan
type." Proc. London Math. Soc., 50, 421-435, 1949.Bressoud,
D. M. Analytic
and Combinatorial Generalizations of the Rogers-Ramanujan Identities. Providence,
RI: Amer. Math. Soc., 1980.Fulman, J. "The Rogers-Ramanujan Identities,
The Finite General Linear Groups, and the Hall-Littlewood Polynomials." Proc.
Amer. Math. Soc.128, 17-25, 1999.Garsia, A. M. and
Milne, S. C. "A Method for Constructing Bijections for Classical Partition
Identities." Proc. Nat. Acad. Sci. USA78, 2026-2028, 1981a.Garsia,
A. M. and Milne, S. C. "A Rogers-Ramanujan Bijection." J.
Combin. Th. Ser. A31, 289-339, 1981b.Guy, R. K. "The
Strong Law of Small Numbers." Amer. Math. Monthly95, 697-712,
1988.Hardy, G. H. Ramanujan:
Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York:
Chelsea, pp. 13 and 90-99, 1999.Hardy, G. H. and Wright, E. M.
"The Rogers-Ramanujan Identities." §19.13 in An
Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon
Press, pp. 290-294, 1979.MacMahon, P. A. Combinatory
Analysis, Vol. 2. New York: Chelsea, pp. 33-36, 1960.Mc
Laughlin, J.; Sills, A. V.; and Zimmer, P. "Dynamic Survey DS15: Rogers-Ramanujan-Slater
Type Identities." Electronic J. Combinatorics, DS15, 1-59, May 31, 2008.
http://www.combinatorics.org/Surveys/ds15.pdf.Paule,
P. "Short and Easy Computer Proofs of the Rogers-Ramanujan Identities and of
Identities of Similar Type." Electronic J. Combinatorics1, No. 1,
R10, 1-9, 1994. http://www.combinatorics.org/Volume_1/Abstracts/v1i1r10.html.Petkovšek,
M.; Wilf, H. S.; and Zeilberger, D. A=B.
Wellesley, MA: A K Peters, p. 117, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.Ramanujan,
S. Problem 584. J. Indian Math. Soc.6, 199-200, 1914.Robinson,
R. M. "Comment to: 'A Motivated Proof of the Rogers-Ramanujan Identities.'
" Amer. Math. Monthly97, 214-215, 1990.Rogers, L. J.
"Second Memoir on the Expansion of Certain Infinite Products." Proc.
London Math. Soc.25, 318-343, 1894.Rogers, L. J. "On
Two Theorems of Combinatory Analysis and Some Allied Identities." Proc. London
Math. Soc.16, 315-336, 1917.Rogers, L. J. "Proof
of Certain Identities in Combinatory Analysis." Proc. Cambridge Philos. Soc.19,
211-214, 1919.Schur, I. "Ein Beitrag zur additiven Zahlentheorie
und zur Theorie der Kettenbrüche." Sitzungsber. Preuss. Akad. Wiss.
Phys.-Math. Klasse, pp. 302-321, 1917.Slater, L. J. "Further
Identities of the Rogers-Ramanujan Type." Proc. London Math. Soc. Ser. 254,
147-167, 1952.Sloane, N. J. A. Sequences A003106/M0261,
A003114/M0266, and A006141/M0260
in "The On-Line Encyclopedia of Integer Sequences."Watson,
G. N. "A New Proof of the Rogers-Ramanujan Identities." J. London
Math. Soc.4, 4-9, 1929.Watson, G. N. "Theorems
Stated by Ramanujan (VII): Theorems on Continued Fractions." J. London Math.
Soc.4, 39-48, 1929.
,
the Rogers-Ramanujan identities are given by (Hardy 1999, pp. 13 and 90),
is a q-Pochhammer
symbol.
when 
. (5) and (6)
specify a sequence of polynomials, indexed by 
, as 
gets larger and larger, the polynomial approximates the series
(◇) more and more closely. Also, taking the limit as 
recovers (◇) after applying the Jacobi
triple product identity on the right hand side. Finally, (5)
and (6) are actually equivalent to each other; (6)
is the "symmetrized" version of (5) using the so-called
Paule symmetrization.
,
1.
with minimal difference 
is equal to the number of partitions into parts of
the forms 
or 
(Hardy 1999, p. 92). The following table gives the
first few values.
(mod 5)
, 
, 
, 
).
).
).
),
but the series representations are double series and therefore not as elegant as
the other identities.
type products.
and 
, there is an "identity" in which the product
side contains 
, so the identity reduces
to 
and therefore is not listed.
).
).
).
. A. Sills worked out a series expansion these
identities, but it was so ugly that he did not publish it (A. Sills, pers. comm.,
Mar. 16, 2005).
