[seqfan] OT: q-binomial-oids (ascii art included)
Joerg Arndt
arndt at jjj.de
Wed Mar 17 18:28:44 CET 2010
[crossposted to seqfan because we have several
people here who might be able to help]
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?
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)} }
[off to the beers, I'll be back tomorrow]
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