Classical Partition Identities and Basic Hypergeometric Series
27
B5. Partitions into distinct parts and finite q-differences
Similarly, let Q` (n|m) be the number of partitions of n into exactly ` distinct
parts with each part ≤ m. Then we have generating functions
X
m (1+`
q 2 ) =
Q` (n|m) qn
(B5.1)
`
n≥0
m
Y
(1 + xqk ) =
k=1
X
Q`(n|m) x` qn
(B5.2)
`,n≥0
whose combination leads us to Euler’s finite q-differences
(x; q)n =
n−1
Y
n
X
`=0
k=0
(1 − xq` ) =
(−1)k
n (k2) k
q x .
k
(B5.3)
Following the second proof of Theorem B3.1, we can check without difficulty
that the shifted Ferrers diagrams of the partitions into `-parts ≤ m are
`+1
unions of the same triangle of length ` enumerated by q( 2 ) and the ordinary
partitions into parts ≤ m − ` with length ≤ ` whose generating function
hmi
reads as the q-binomial coefficient
. The product of them gives the
`
generating function for {Q`(n|m)}n .
The second formula is a particular case of (B1.2b). Its combination with
the univariate generating function just proved leads us to the following:
m X
m
X
X
1+`
n `
` m ( 2 )
(−qx; q)m =
q
.
Q`(n|m) q x =
x
`
`=0 n≥0
`=0
Replacing x by −x/q in the above, we get Euler’s q-difference formula:
m
X
` m (2` ) `
(x; q)m =
q x.
(−1)
`
`=0
Remark The last