I can find the following in the literature:

___n-1 M-k | | (1-q ) ___M-1 n \~~ M | |k=0 n n (n-1)/2 | | (1+x q ) = > -------------- x q | |n=0 /__ n=0 ___n k | | (1-q ) | |k=1

Now I came up with

___n-1 M-k | | (1-q ) 1 \~~ M | |k=0 n n (n-1) -------------- = > ------------------------------- x q ___M-1 n /__ n=0 ___n-1 k ___n-1 k | | (1-x q ) | | (1-q q ) | | (1-x q ) | |n=0 | |k=0 | |k=0

This is certainly known. Can anyone point out where this is given? Herr Gosper? LaTeX sources are \prod_{n=0}^{M-1}{(1+x\,q^n)} & = & \sum_{n=0}^{M}{ \frac{\prod_{k=0}^{n-1}{(1-q^{M-k})}} {\prod_{k=1}^{n}{(1-q^k)}} \, x^n \, q^{n\,(n-1)/2} } and \frac{1}{\prod_{n=0}^{M-1}{(1-x\,q^n)}} & = & \sum_{n=0}^{M}{ \frac{\prod_{k=0}^{n-1}{(1-q^{M-k})}} {\prod_{k=0}^{n-1}{(1-q\,q^k)} \, \prod_{k=0}^{n-1}{(1-x\,q^k)}} \, x^n \, q^{n\,(n-1)} } Modulo some changes of variable (e.g. to obviate x), and some product massaging, that's the q-Chu-Vandermonde sum, Basic Hypergeometric Series, Appendix II, II.7 . This came up here 8 nov 05, when I derived the known result:

the probability that an nxn bitmatrix will have (mod 2) rank = k is

2 n - k + 1 k + 1 (n - k) (q ; q) (q ; q) q k n - k

P(n,k) := -------------------------------------------, (q; q) n - k

if the entries are 1 with probability p = 1-q .

More generally, for an m x n matrix, m - k + 1 k + 1 (m - k) (n - k) (q ; q) (q ; q) q k n - k P(m, n, k) = --------------------------------------------------. (q; q) n - k --rwg