# [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]