Andrews-Schur Identity

The Andrews-Schur identity states

(1)

where  is a q-binomial coefficient and  is a q-bracket. It is a polynomial identity for , 1 which implies the Rogers-Ramanujan identities by taking  and applying the Jacobi triple product identity.

The limit as  of the identity in (1) is

(2)

A variant of the identity is

(3)

where the symbol  in the sum limits is the floor function (Paule 1994). A related identity is given by

(4)

for , 1 (Paule 1994). For , equation (3) becomes

(5)

REFERENCES:

Andrews, G. E. "A Polynomial Identity which Implies the Rogers-Ramanujan Identities." Scripta Math. 28, 297-305, 1970.

Paule, P. "Short and Easy Computer Proofs of the Rogers-Ramanujan Identities and of Identities of Similar Type." Electronic J. Combinatorics 1, No. 1, R10, 1-9, 1994. http://www.combinatorics.org/Volume_1/Abstracts/v1i1r10.html.