* Fred lunnon <fred.lunnon@gmail.com> [Mar 19. 2010 15:20]:
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 x q ) <--= HERE | |n=0 | |k=0 | |k=0
Thanks for spotting! An 'x' has to be inserted in the rightmost lower factor (done in the ascii art above). This was a error with TeX-ing up from pari/gp code: L(M,a,b)=prod(n=0,M-1,(1-a*x*q^n)/(1-b*x*q^n)) R(M,a,b)=sum(n=0,M, prod(k=0,n-1,1-q^(M-k)) / prod(k=1,n,1-q^k) * q^(n*(n-1)/2) * x^n * prod(k=0,n-1,b*q^k-a) / prod(k=0,n-1,1-b*x*q^k) ) \\ check first few: { for(M=0,5, f=L(M,a,b); g=R(M,a,b); print([M,f-g]); ); } \\ this prints [0, 0] [1, 0] [2, 0] [3, 0] [4, 0] [5, 0]