* Bill Gosper <billgosper@gmail.com> [Feb 28. 2012 08:14]:

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Dawk! And better still as one Lambert series!

Joerg>(first:)

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}] == (second:) Sum[(-x)^n/(1 - q^(2*n)), {n, 1, Infinity}] + (1/2)*x*Sum[q^n/(1 + q^n*x), {n, 0, Infinity}] == (third:) (-(1/2))*x*Sum[(-q)^n/(1 + q^n*x), {n, 0, Infinity}]

Prettier still (except for taking x -> +or-1) (fourth:) (-(1/2))*x*Sum[(-q)^n/(1 + q^n*x), {n, 0, Infinity}] == (1/2)*Sum[(-x)^j/(1 + q^j), {j, 1, Infinity}] These Lambert transformations are all the same easy trick, which I'm not stating because I've already asked Funster Neil (who owes us his sliding block report) to figure it out for tomorrow. Joerg> The equality (first) == (third) appears to be a specialization of my relation (7): \begin{equation}\label{rel:lambert-qxt-alt} % sum(n=0,N, t^n/(1-x*q^n) ); \sum_{n\geq{}0}{ \frac{t^n}{1-x\,q^{n}} } = % % sum(n=0,N, (qbin(q,q,n)) / (qbin(x,q,n+1)*qbin(t,q,n+1)) * (-x*t)^n * q^((n^2-n)/2) ); \sum_{n\geq{}0}{ \frac{ (q;q)_n }{ (x;q)_{n+1} \, (t;q)_{n+1} } \, (x\,t)^n\, q^{(n^2-n)/2} } \end{equation} Joerg>The equality (second) == (third) resembles the "well known" sum(n>=1, x*q^n/(1-x*q^n) ) == -x/(1-x) + sum(n>=1, x^n/(1-q^n) ) (but I cannot morph this one into your's right now, replacing x by -x may be a good start).

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This whole limit puzzle arose from the false hope that a->-1 would nicely simplify the lhs of (QPochhammer[a, q]*QPochhammer[(a*q)/x, q]*QPochhammer[a*x, q]*         (Sum[(a^n*QPochhammer[x/a, q, n])/QPochhammer[a*x, q, n], {n,         0, Infinity}] + (a*(x + q)*          Sum[(QPochhammer[-a, q, n]*q^n*QPochhammer[a^2*q, q^2, n])/                     (QPochhammer[a, q, n + 1]*QPochhammer[(a*q)/x, q, n + 1]*                        QPochhammer[a*x, q, n]), {n, 0, Infinity}])/x))/      (QPochhammer[-a, q]*QPochhammer[a^2*q, q^2]) ==  Sum[(-1)^n*Sqrt[q]^((n - 1)*n)*x^n,  {n, 0, Infinity}] giving a "unilateral triple product identity". --rwg Re Provide and Require Macsyma was written in Lisp, but its SETUP_AUTOLOAD may predate Lisp's equivalent functionality.  However,  the autoloading occurs at runtime, not compile time.