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

18 S. O. WARNAARThen, for L, M ∈ Z + ,∑ ∑(7.6) ɛ(σ)q φ p,k,σT ′′ (L, M, pk − σ + ρ, (p − 1)k − σ + ρ)σ∈S n|k|=0= ∑ µwith sum over η as in (7.1) and η given by[ ][ ]q 1 2 µ(C−1 ⊗C)µ M + nL −∑b C−1 1,b µ(b) 1 µ + ηnLµη = nL(C −1 e 1 ⊗ e 1 ) − (C −1 ⊗ C)µ.The left-hand side of this identity coincides with the generating function intheorem 3 of [17] with r → n, λ → (L n ), µ → (0 n ), d → p − n, α → (0 p−n ),β → (0, . . . , 0, n − p + 1), a i → M and b i → 0. For p = n + 1 the above isidentity (6.2) of [17].We note that (7.4), (7.5) and (7.6) are consistent. Specifically, (7.4) withM 2 → ∞ and (7.6) with L → ∞ coincide (identifying M 1 with M), and (7.5)with L = ((n−1)L 1 , . . . , 2L 1 , L 1 ) and (7.6) with M → ∞ coincide (identifyingL 1 and L).The three above conjectures strongly suggest the existence of a unifyingidentity of the form∑ ∑(7.7) ɛ(σ)q φ p,k,σT (L, M, pk − σ + ρ, (p − 1)k − σ + ρ)σ∈S n|k|=0= ∑ µn−1q 1 2 µ(C−1 ⊗C)µ∏( [ M a + (CL) a − ∑ b C−1a=1(CL) aa,b µ(b) 1] ) [ ] µ + ηwith sum over µ such that (7.1) holds, η given by (7.2) and L, M ∈ Z n−1+ . Thegeneralized supernomial T (L, M, k, k ′ ) (where L, M ∈ Z n−1 , k, k ′ ∈ Z n and|k| = |k ′ | = 0) must satisfy the following consistency conditions:lim T (L, (M 1 , 0 n−3 , M 2 ), k, k ′ ) = T ((M 1 , M 2 ), k)(q) M1 +MCL→(∞ n−1 2)lim T (L, M, k, k ′ ) = T ′ (L, k)M→(∞ n−1 )T (((n − 1)L 1 , . . . , 2L 1 , L 1 ), (M 1 , 0 n−2 ), k, k ′ ) = T ′′ (L 1 , M 1 , k, k ′ ).(The first condition applies when n ≥ 3 only.) A further restriction on thepossible form of T is obtained by observing that the right-hand side is, up toa factor q MCL , invariant under the change q → 1/q, so thatT (L, M, k, k ′ ; 1/q)T (L, M, k, k ′ ; q)= q −MCL+∑ ni=1 k ik ′ i .Despite these strong restrictions on T (especially when n = 3) we have notsucceeded in finding a closed form expression when n ≥ 3. For n = 2 theµ