On Tue, Mar 6, 2012 at 2:12 AM, Bill Gosper <billgosper@gmail.com> wrote:
in http://gosper.org/fst.pdf, namely
((-1 + QPochhammer[a, p])*QPochhammer[q, q])/a == Sum[(q^((1/2)*n*(1 + n))* Product[1 - a*p^k + c*p^k*q^k - q^(k - n), {k, 0, Infinity}])/ ((-1)^n*((a - c*q^n)*QPochhammer[q, q, n])), {n, 0, Infinity}]
eluded our attempts to rederive it, despite the hints near the end of the paper. Those led me instead to
((-1 + QPochhammer[a, p])*QPochhammer[q, q])/a == Sum[-(((-1)^n*q^((1/2)*n*(3 + n))* Product[1 + (-1 + a*p^k)*q^(k - n) + (a*c*p^k)/q^n, {k, 0, Infinity}])/(a*(c + q^n)* QPochhammer[q, q, n])), {n, 0, Infinity}]
Trying to guess how far off the mainstream this result lurks, I put p=q, c=1/a: (1 - QPochhammer[a, q])*QPochhammer[q, q] == Sum[((-1)^n*q^((1/2)*n*(3 + n))*QPochhammer[-a/q^n, q^2])/ ((1/a + q^n)*QPochhammer[q, q, n]), {n, 0, Infinity}] which is still middling strange. BtW, Identity (5) is generalized to Bessel functions in projecteuclid.org/euclid.ijm/1255987146 whose results are derived completely differently in www.ams.org/proc/1993-117.../S0002-9939-1993-1116276-8.pdf The former reference suggests that Jet Wimp may have anticipated nonlocal derangement. (I'm not sure when I found it.) I suspect everyone is rushing off doing Algebraic Foobology just to avoid rediscovering things. --rwg MIDPENINSULA INDIAN PLUMES
Series sez they both work! The summands are utterly unlike, even for n = 0. (Idiot Mma takes ~minutes to plug n=0 into the ratio of the summands! No sums, just products.)
Changing c->-a/c in the former, even though unmentioned in the paper,
((-1 + QPochhammer[a, p])*QPochhammer[q, q])/a == Sum[((-1)^n*c*q^((1/2)*n*(3 + n))* Product[1 + (-1 + a*p^k)*q^(k - n) - (a^2*p^k)/(q^n*c), {k, 0, Infinity}])/(a*(a - c*q^n)* QPochhammer[q, q, n]), {n, 0, Infinity}]
did zilch to reconcile the summands. So, I don't know where the bleep (4) came from, but I've regained some facility with nonlocal derangement, and now we have (4a) and (4b). --rwg