* Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> [Feb 27. 2012 14:27]:> On Sun, Feb 26, 2012 at 12:32 AM, Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> wrote:> > > (-1/2 + Sum[(a^n*QPochhammer[x/a, q, n])/QPochhammer[a*x, q, n], {n, 0,> > Infinity}])/> > (1 + a)> >> > as a -> -1+ . (In terms of q-hypergeometric(s) in x and q).> > > Here's a much better answer as two Lambert series:> > Limit[(-1/2 +> Sum[(a^n*QPochhammer[x/a, q, n])/QPochhammer[a*x, q, n],> {n, 0, Infinity}])/(1 + a), a -> -1, Direction -> -1] ==> 1/4 + 2*Sum[(-x)^n/(1 - q^(2*n)), {n, 1, Infinity}] +> x*Sum[q^n/(1 + q^n*x), {n, 0, Infinity}] Dawk! And better still as one Lambert series! Sum[((-1)^k*q^((1/2)*(-1 + k)*k)*x^k*QPochhammer[q, q, -1 + k])/ (QPochhammer[-1, q, 1 + k]*QPochhammer[-x, q, k]), {k, 1, Infinity}] == Sum[(-x)^n/(1 - q^(2*n)), {n, 1, Infinity}] + (1/2)*x*Sum[q^n/(1 + q^n*x), {n, 0, Infinity}] == (-(1/2))*x*Sum[(-q)^n/(1 + q^n*x), {n, 0, Infinity}] rwg> This came from combining Heine's first and second> transformations http://dlmf.nist.gov/17.6 (17.6.6, 17.6.7) to get> QHypergeometricPFQ[{a, b}, {c}, q, z]==> QHypergeometricPFQ[{c/b, z}, {a z}, q, b] *> QPochhammer[b, q] QPochhammer[a z, q])/> (QPochhammer[c, q] QPochhammer[z, q])> > This transforms the sum in the limitand into one with numerator> parameter a^2, which means all its terms after the 0th will contain> a removable factor of 1-a^2 = (1-a)(1+a), and the 0th minus 1/2> and then divided by a+1 contributes one of the Lambert sums> as a -> -1.> > [...] Joerg>Not sure this is pertinent here, but see relation (3.1) on p.604 of %\bibitem{Chan}{Song Heng Chan: % {Generalized Lambert Series Identities}, % Proceedings of the London Mathematical Society, vol.91, no.3, pp.598-622, (2005). %}%% rel.(3.1): expression for bilateral Lambert series "Subscription required." Joerg>Can send the pdf in case the paper isn't open access. That would be sweet. --rwg Joerg> IIRC there is something similar in Fine, but can only point to his relation (18.4) on p.20 right now.