On 2015-08-25 00:59, Joerg Arndt wrote:
* rwg <rwg@sdf.org> [Aug 25. 2015 08:14]:
[...]
Ahh, here's our problem-- I do not accept Lambert series as closed form!
But why is then Theta_2 acceptable? 'Cause Whittaker & Watson legitimize it.
(no idea what these QPolyGamma thingies are)
In[451]:= D[Log[QGamma[x, q]], x] Out[451]= QPolyGamma[0, x, q]
To do without them, http://gosper.org/recipfib.pdf [1] [2 [1]]
In[147]:= D[Log[QPochhammer[q, q]], q]
Out[147]= (-((QPochhammer[q, q]*(Log[1 - q] + QPolyGamma[0, 1, q]))/(q*Log[q])) + Derivative[0, 1][QPochhammer][q, q])/ QPochhammer[q, q]
Sum[1/Fibonacci[n], {n, Infinity}] == (1/4)*Sqrt[5]* ((-4*QPolyGamma[0, 1, 1/GoldenRatio^2] + 2*QPolyGamma[0, 1, 1/GoldenRatio^4] + Log[5])/(2*Log[GoldenRatio]) + EllipticTheta[2, 0, 1/GoldenRatio^2]^2)
N[List @@ %, 49]
{3.359885666243177553172011302918927179688905133732, 3.359885666243177553172011302918927179688905133732}
BtW, there's a very simple path-invariant q matrix system for converting generalized Lamberts to theta convergence.
Let's us see! http://gosper.org/lambser.png [4] --rwg
--rwg
Links: ------ [1] http://arxiv.org/abs/1202.6525 [2] [2] http://gosper.org/recipfib.pdf [1] _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun [3]
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Links: ------ [1] http://gosper.org/recipfib.pdf [2] http://arxiv.org/abs/1202.6525 [3] https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun [4] http://gosper.org/lambser.png