* Bill Gosper <billgosper@gmail.com> [Feb 11. 2012 15:10]:
[...] Aggrieved at not finding Joerg's Theta-convergent, Pochhammer-free identity, I exhaustively searched all computationally feasible coordinate changes of this 4D system. The winner was i=t-1, j=-1, leaving the extremely simple 2D system [ n + 2 k + i - 1 ] [ n + 2 k + i 1 - q ] [ q ----------------------------- ] [km(k, n) := [ k + i n + k - 1 ], [ (1 - q ) (1 - q ) ] [ ] [ 0 1 ]
[ k + i 1 ] [ q -------------- ] nm(k, n) := [ n + k - 1 ]] [ 1 - q ] [ ] [ 0 1 ] (the specialization of j to -1 permits "sidestepping" to almost pure sum notation, ruling out Pochhammers from the contour.) Running these through MProd, Julian's new matrix product to sum converter,
{MProd[{{q^(i + 1), 1/(1 - q^n)}, {0, 1}}, {n, x, Infinity}] , MProd[{{0, -(1/(q^(k + i) - 1))}, {0, 1}}, {k, 1, Infinity}]} -> {MProd[{{q^(x + 2*k + i), (1 - q^(x + 2*k + i - 1))/((1 - q^(k + i))*(1 - q^(x + k - 1)))}, {0, 1}}, {k, 1, Infinity}] , {{0, 1}, {0, 1}}}
elicited from Mma a bunch of bogus nonconvergence complaints, then a bunch of infectious and gratuitous "Indeterminate"s, and ultimately an utterly useless and incorrect "False". Changing Equal to Rule and Dot to List, and then changing them back: In[101]:= Dot @@ # & /@ (% /. Indeterminate -> 0) /. Rule -> Equal
Out[101]= {{0, Sum[(q^(1 + i))^(k39 - x)/(1 - q^k39), {k39, x, Infinity}]}, {0, 1}} == {{0, Sum[(q^((-1 + k43)*(i + k43 + x))*(1 - q^(-1 + i + 2*k43 + x)))/((1 - q^(i + k43))*(1 - q^(-1 + k43 + x))), {k43, 1, Infinity}]}, {0, 1}}
At last! --rwg
OK, here is what I get: jj: k39, k43 |--> k { {0, Sum[(q^(1 + i))^(k - x)/(1 - q^k), {k, x,Infinity}]}, {0, 1} } == { {0, Sum[(q^((-1 + k)*(i + k + x))*(1 - q^(-1 + i + 2*k + x))) / ((1 - q^(i + k))*(1 - q^(-1 + k + x))), {k, 1, Infinity}]}, {0, 1} } jj: x |--> 1 { {0, Sum[(q^(1 + i))^(k - 1)/(1 - q^k), {k, 1, Infinity}]}, {0, 1} } == { {0, Sum[(q^((-1 + k)*(i + k + 1))*(1 - q^(i + 2*k))) / ((1 - q^(i + k))*(1 - q^(k))), {k, 1, Infinity}]}, {0, 1} } jj: i |--> 0 { {0, Sum[q^(k-1)/(1 - q^k), {k, 1, Infinity}]}, {0, 1} } == { {0, Sum[ ( q^((k-1)*(k+1)) * (1 - q^(2*k)) ) / ( (1 - q^(k)) * (1 - q^(k)) ), {k, 1, Infinity}]}, {0, 1} } And this is Clausen's \sum_{n\geq{}1}{ \frac{q^n}{1-q^{n}} } = \sum_{n\geq{}1}{ \frac{1+q^{n}}{1-q^{n}} \, q^{n^2} } (given on p.95 of \bibitem{Clausen}{Thomas Clausen: {Beitrag zur Theorie der Reihen}, Journal f\"{u}r die reine und angewandte Mathematik, vol.3, pp.92-95, (1828).} %% http://www.digizeitschriften.de/main/dms/toc/?PPN=PPN243919689_0003 ) Will mention this matrix magic in the newly launched section. ... but now it's Beer o'clock thanks, jj
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