* Joerg Arndt <arndt@jjj.de> [Feb 22. 2012 11:19]:
* Bill Gosper <billgosper@gmail.com> [Feb 18. 2012 07:42]:
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Linearly combining other results gives a thetalike gfcn for the reciprocal pochhammers 1/(x;q)_k in terms of the gfcn for the unreciprocated pochhammers (t;q)_k !
Sum[x^k*QPochhammer[t, q, k], {k, 0, Infinity}] == Sum[(q^(-(k/2) + k^2/2)*t^k*x^k)/((-1)^k*QPochhammer[x, q, 1 + k]), {k, 0, Infinity}] --rwg
I'll check this one next.
Done. \\ Your \\ default(echo,1); \\ > > Sum[x^k*QPochhammer[t, q, k], {k, 0, Infinity}] == \\ > > Sum[(q^(-(k/2) + k^2/2)*t^k*x^k)/((-1)^k*QPochhammer[x, q, 1 + k]), {k, 0, Infinity}] \\ is, in pari/gp: t1 = sum(n=0,N, x^n * qbin(t,q,n) ) t2 = sum(n=0,N, ( (-x*t)^n * q^((n^2-n)/2) ) / qbin(x,q,n+1) ) t1-t2 \\ == zero \\ ... which is Fine's relation (6.1): t3 = sum(n=0,N, x^n * qbin(q*t,q,n) ) t4 = sum(n=0,N, ( (-x*t)^n * q^((n^2+n)/2) ) / qbin(x,q,n+1) ) t3-t4 \\ == zero
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