[math-fun] Reciprocal fib sum
Borwein*2, Exercise 3.7.10 (at least my old edition): "A remarkable elementary result is Sum[1/(Fibonacci[1 + 2*k] + Fibonacci[1 + 2*n]), {n, 0, Infinity}] == (Sqrt[5]*(1 + 2*k))/ (2*Fibonacci[1 + 2*k]) [. . .] A related formula is [...]" which is just the k=3 case of the first formula, but with a different, and correct rhs! I.e., the first formula seems completely wrong, Empirically, I get Sum[1/(Fibonacci[1 + 2*k] + Fibonacci[1 + 2*n]), {n, 0, Infinity}] == ((1 + GoldenRatio^(2 + 4*k))* (1 + 2*k))/(2*(-1 + GoldenRatio^(2 + 4*k))* Fibonacci[1 + 2*k]) The rhs might have a slick reexpression in fibonaccis. --rwg (Recall our old sum 1/fib(n) saga, where B&B seem to cop out and give only a Lambert series, but reading between the lines, there's a hairy closed form in terms of d/dq log qpochhammers.)
* Bill Gosper <billgosper@gmail.com> [Aug 24. 2015 16:52]:
Borwein*2, Exercise 3.7.10 (at least my old edition):
I do not think that there was another than the first edition. When I talked to Jon Borwein, suggesting a new edition would be a good thing, he essentially said "never gonna happen".
[...]
(Recall our old sum 1/fib(n) saga, where B&B seem to cop out and give only a Lambert series, but reading between the lines, there's a hairy closed form in terms of d/dq log qpochhammers.)
But there is a nice (theta convergence) "closed form", see section 2 of http://arxiv.org/abs/1202.6525 8^) The thing is more general than for 1/fib, it covers positive discriminants (of the recurrence). I have not attempted to crack negative discriminants, though that would appear worthwhile (I'd start looking at the Binet forms and try to carry the logic from positive discriminants over, looks doable). Best regards, jj
[...]
On 2015-08-24 09:05, Joerg Arndt wrote:
* Bill Gosper <billgosper@gmail.com> [Aug 24. 2015 16:52]:
Borwein*2, Exercise 3.7.10 (at least my old edition):
I do not think that there was another than the first edition. When I talked to Jon Borwein, suggesting a new edition would be a good thing, he essentially said "never gonna happen".
[...]
(Recall our old sum 1/fib(n) saga, where B&B seem to cop out and give only a Lambert series, but reading between the lines, there's a hairy closed form in terms of d/dq log qpochhammers.)
But there is a nice (theta convergence) "closed form", see section 2 of http://arxiv.org/abs/1202.6525 [1] 8^)
The thing is more general than for 1/fib, it covers positive discriminants (of the recurrence). I have not attempted to crack negative discriminants, though that would appear worthwhile (I'd start looking at the Binet forms and try to carry the logic from positive discriminants over, looks doable).
Best regards, jj
[...]
Ahh, here's our problem-- I do not accept Lambert series as closed form! To do without them, http://gosper.org/recipfib.pdf [2] 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. --rwg Links: ------ [1] http://arxiv.org/abs/1202.6525 [2] http://gosper.org/recipfib.pdf
* 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? (no idea what these QPolyGamma thingies are)
To do without them, http://gosper.org/recipfib.pdf [2]
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!
--rwg
Links: ------ [1] http://arxiv.org/abs/1202.6525 [2] http://gosper.org/recipfib.pdf _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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]
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun [3]
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
participants (3)
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Bill Gosper -
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rwg