Curious asymmetry --- (b q - a) on top, (1 - b q) on bottom --- have you checked there's no typo here? WFL On 3/19/10, Joerg Arndt <arndt@jjj.de> wrote:
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