Rogers-Ramanujan Identities

For $\vert q\vert<1$ and using the Notation of the Ramanujan Theta Function, the Rogers-Ramanujan identities are

$${f(-q^5)\over f(-q,-q^4)} = \sum_{k=0}^\infty {q^{k^2}\over (q)_k}$$ (1)

$${f(-q^5)\over f(-q^2,-q^3)} = \sum_{k=0}^\infty {q^{k(k+1)}\over (q)_k},$$ (2)

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

$$1+{q\over 1-q}+{q^4\over (1-q)(1-q^2)}+{q^9\over (1-q)(1-q^2... ...q^3)}+\ldots = {1\over (1-q)(1-q^6)\cdots(1-q^4)(1-q^9)\cdots}$$ (3)

$$1+{q^2\over 1-q}+{q^6\over (1-q)(1-q^2)}+{q^{12}\over (1-q)(... ...)}+\ldots = {1\over (1-q^2)(1-q^7)\cdots(1-q^3)(1-q^8)\cdots}.$$ (4)

The identities can also be written succinctly as

$$1+\sum_{k=1}^\infty {q^{k^2+ak}\over (1-q)(1-q^2)\cdots(1-q^k)} =\prod_{j=0}^\infty {1\over (1-q^{5j+a+1})(1-q^{5j-a+4})},$$ (5)

where $a=0$, 1.

Other forms of the Rogers-Ramanujan identities include

$$\sum_k {q^{k^2}\over(q;q)_k(q;q)_{n-k}}=\sum_k{(-1)^kq^{(5k^2-k)/2}\over (q;q)_{n-k}(q;q)_{n+k}}$$ (6)

and

$$\sum_k {2q^{k^2}\over(q;q)_k(q;q)_{n-k}}=\sum_k {(-1)^k(1+q^k)q^{(5k^2-k)/2}\over(q;q)_{n-k}(q;q)_{n+k}}$$ (7)

(Petkovsek et al. 1996).

See also Andrews-Schur Identity

References

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

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. 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.