Euler's partition theorem and the combinatorics of -sequences

Define a q-analog of the l-nomial:
a n
(l) (q) = un l − v l nqn
u l − v l q ;
∆(l) n (q) = ul n + v l n qn
[ n
k
] (l)
q
= a(l) n (q) a (l)
n−1
(q) · · · a(l)
n−k+1 (q)
a (l) (q) a(l) (q) · · · a(l)
1 (q) .
k
k−1
Then
[ n
k
] (l)
q
= q k [ n − 2
k
] (l)
q
[ ]
+ ∆ (l) n − 2 (l)
n−1 (q) k − 1
q
[ ] n − 2 (
+ q n−k k − 2
q
and
n∑
[ n
k
k=0
] (l)
q
q k(k+1)/2 z k = α n (q, z)
⌊n/2⌋
∏
i=1
(1 + zq i ∆ (l)
n−2i+1 (q) + z2 q n+1 ).
where α n (q, z) = 1 if n is even; (1 + zq (n+1)/2 ) if n is odd.