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Date: 28-7-2019
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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.
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