* Bill Gosper <billgosper@gmail.com> [Feb 18. 2012 07:42]:
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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}]
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}