[seqfan] Re: OT: q-binomial-oids (ascii art included)
Joerg Arndt
arndt at jjj.de
Fri Mar 19 11:11:22 CET 2010
Jacobi! (this was hard to find)
> The relations are respectively the special cases (a,b)=(-1,0) and
> (a,b)=(0,1) of an identity due to Jacobi:
>
> ___M-1 n ___n-1 M-k ___n-1 k
> | | (1-a x q ) | | (1-q ) | | (b q -a)
> | |n=0 \~~ M | |k=0 | |k=0 n n (n-1)/2
> -----------------= > ----------------------------------- x q
> ___M-1 n /__ n=0 ___n-1 k ___n-1 k
> | | (1-b x q ) | | (1-q ) | | (1-b q )
> | |n=0 | |k=0 | |k=0
>
To be found on p.795 of
W. P. Johnson:
{How Cauchy Missed Ramanujan's ${}_1\psi_1$ Summation},
American Mathematical Monthly,
vol.111, no.9, pp.791-800, November-2004
* Joerg Arndt <arndt at jjj.de> [Mar 18. 2010 08:02]:
> [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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