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Andrews-Schur Identity

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The Andrews-Schur identity states

sum_(k=0)^nq^(k^2+ak)[2n-k+a; k]_q =sum_(k=-infty)^inftyq^(10k^2+(4a-1)k)[2n+2a+2; n-5k]_q([10k+2a+2]_q)/([2n+2a+2]_q),
(1)

where [n; m]_q is a q-binomial coefficient and [n]_q is a q-bracket. It is a polynomial identity for a=0, 1 which implies the Rogers-Ramanujan identities by taking n->infty and applying the Jacobi triple product identity.

The limit as n->infty of the identity in (1) is

1/((q^(4-a);q^5)_infty(q^(a+1);q^5)_infty).
(2)

A variant of the identity is

sum_(k=-|_a/2_|)^nq^(k^2+2ak)[n+k+a; n-k]_q =sum_(-|_(n+2a+2)/5_|)^(|_n/5_|)q^(15k^2+(6a+1)k)[2n+2a+2; n-5k]_q([10k+2a+2]_q)/([2n+2a+2]_q),
(3)

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

sum_(k=0)^infty(q^(k^2+2ak))/((q;q)_(2k+a))=product_(j=0)^infty1/((1-q^(2j+1))(1-q^(20j+4a+4))(1-q^(20j-4a+16)))
(4)

for a=0, 1 (Paule 1994). For q=1, equation (3) becomes

sum_(k=-|_a/2_|)^n(n+k+a; n-k)=sum_(k=-|_(n+2a+2)/5_|)^(|_n/5_|)(2n+2a+2; n-5k)(5k+a+1)/(n+a+1).
(5)

See also

Rogers-Ramanujan Identities

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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." Elec. J. Combin. 1, R10, 1994. https://doi.org/10.37236/1190.

Referenced on Wolfram|Alpha

Andrews-Schur Identity

Cite this as:

Weisstein, Eric W. "Andrews-Schur Identity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Andrews-SchurIdentity.html

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