SUPERNOMIAL COEFFICIENTS, BAILEY'S LEMMA AND ROGERS ...

10 S. O. WARNAARIt remains to find identities for moduli congruent to 0 (mod 3). What isneeded now is the A 2 analogue of identity (3.4) provided by Gessel and Krattenthaler(equation (6.18) of [17]).Proposition 4.5. For L ∈ Z 2 +,(4.4)∑ ∑ɛ(σ)q 1 ∑ 32 i=1 (3k i−σ i +i) 2 T (L, 3k − σ + ρ) =(q3 ; q 3 ) |L|.(q 3 ; q 3 ) L (q) 2 σ∈S 3 |L||k|=0Applying theorem 3.2 this readily gives∑ ∑ɛ(σ)q l ∑ 32 i=1 (3k i−σ i +i) 2 T (L, 3k − σ + ρ)σ∈S 3|k|=0=∑n 1 ,...,n l−1 ∈Z 2 +q 1 ∑ l−12 j=1 N jCN j(q 3 ; q 3 ) |nl−1 |.(q) L−N1 (q) n1 · · · (q) nl−2 (q 3 ; q 3 ) nl−1 (q) 2 |n l−1 |When L 1 , L 2 approach infinity this yields our final Rogers–Ramanujan-typetheorem (theorem 5.4 of [9] with i = k).Theorem 4.6. For |q| < 1, k ≥ 2 and N j = n j + · · · + n k−1 ,∑q 1 2∑ k−1j=1 N jCN j(q 3 ; q 3 ) |nk−1 |n 1 ,...,n k−1 ∈Z 2 +(q) n1 · · · (q) nk−2 (q 3 ; q 3 ) nk−1 (q) 2 |n k−1 |= (qk , q k , q k , q 2k , q 2k , q 2k , q 3k , q 3k ; q 3k ) ∞.(q) 3 ∞5. Reduction and inversionSo far, we have used the A 1 and A 2 summations (2.2), (3.9) and (3.10) toderive complicated identities out of simpler ones. Of, course, when given acomplicated identity, it is of interest to know whether this identity is reducibleto a simpler one. That is, whether, iteration of some yet unknown simpleridentity produces the complicated identity. The answer to this question iseasily given, and in the case of A 1 the following result holds [7] (which actuallyis the q → 1/q version of (2.2))(5.1)M∑L=0(−1) M−L q (M−L 2 )−M 2(q) M−LS(L, r) = q −r2 S(M, r).Iterating this identity using the invariance property (2.2) impliesN∑q M 2M∑(q) N−MM=0L=0(−1) M−L q (M−L 2 )−M 2(q) M−LS(L, r) = S(N, r).