joerg> * Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> [Feb 18. 2012 07:42]:> [...]

which is somewhat less peculiar when properly simplified:> > Sum[t^n/QPochhammer[x, q, 1 + n], {n, 0, Infinity}] ==> Sum[(q^j*QPochhammer[q^(1 + j), q])/(QPochhammer[q^j*t, q]*> QPochhammer[q^j*x, q]), {j, 0, Infinity}]> DLMF 17.6.12 http://dlmf.nist.gov/17.6 --rwg

joerg>From the equation I get ( N := \infty ) qbin(t,q,N) * qbin(x,q,N) / qbin(q,q,N) * sum(n=0,N, t^n / qbin(x,q,n+1) ) == sum(j=0,N, ( q^j * qbin(t,q,j) * qbin(x,q,j) ) / ( qbin(q,q,j) ) ) Now the second one is invariant against exchange of x and t, and so is the factor of the first one preceding the sum. Hence we have: sum(n=0,N, t^n / qbin(x,q,n+1) ) == sum(n=0,N, x^n / qbin(t,q,n+1) ) Now this a nice counterpart of (my/Osler's) sum(n=0,N, t^n / (1 - x*q^n) ) == sum(n=0,N, x^n / (1 - t*q^n) )

equivalently> QHypergeometricPFQ[{q, 0}, {x}, q, t] ==> (QHypergeometricPFQ[{t, x/q}, {0}, q, q]*QPochhammer[q, q])/> (QPochhammer[t, q]*QPochhammer[x, q])

Which may well be in http://dlmf.nist.gov/ If so, can you point to it?

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So now there's a \section{News from planet Gosper}