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@jjj.de> [Mar 17. 2010 18:22]:
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)} }
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