On 2014-08-29 06:30, Joerg Arndt wrote:

* Bill Gosper <billgosper@gmail.com> [Aug 29. 2014 14:30]:

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Choosing the simplest alternatives in http://mathworld.wolfram.com/ReciprocalFibonacciConstant.html leads to the remarkably simple http://gosper.org/recipfib.pdf .

I was not aware of this. Btw. those \Gamma^{(1,0)} and \Psi^{(0)} should be defined. Agreed.

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Jörg, did you know about this all along? (My Borwein&Borwein is still AWOL.)

Thanks for the peek! Just as I remembered, they give only a Lambert series, which is practically a no-op.

All I know is in http://arxiv.org/abs/1202.6525 Summation only for first terms 0 and 1, and determinant > 0 (I did not find how to treat det < 0 IIRC).

Was aimed at Fib. Quart., but gave up after finding main result in an older paper (as now remarked).

Pari code at http://jjj.de/pari/ (file lambert.gpi, at end of file)

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Wikipedia still has two articles maintaining no closed form.

No closed form with http://en.wikipedia.org/wiki/Reciprocal_Fibonacci_constant But two (in Theta-nulls) with http://en.wikipedia.org/wiki/Fibonacci_number#Reciprocal_sums

No, they just have 1/Fib[2n+1] and 1/Fib[n]^2, etc, like your paper. "No closed formula for the reciprocal Fibonacci constant <https://en.wikipedia.org/wiki/Reciprocal_Fibonacci_constant> [image: \psi = \sum_{k=1}^{\infty} \frac{1}{F_k} = 3.359885666243 \dots] is known,..."

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Are they really stodgy with their definition, or just uninformed?

Not sure what you are aiming at.

Correcting the claim of no closed form. The talk page agonizes needlessly over whether 𝜗 is closed form, but the main Fib article uses it shamelessly for sum 1/Fib[2n+1] etc.

Certainly those theta formulas should appear on the page Reciprocal_Fibonacci_constant as well!

With credit to whom?? I followed a couple of Weisstein's leads and found the q-Gamma' formula nowhere. I'm betting he found it himself, quietly observing that the log derivative of the q-Gamma infinite product is a Lambert series. This is almost obvious, except that Mathematica makes it almost impossible to automatically turn log(product) into sum(logs). Maybe he secretly used Macsyma!-)

From the 23 May discussion,

----------- Jörg>Have you checked Borwein/Borwein "Pi and the AGM"? pp.91-101 might make you happy. rwg>Sadly not)-: They denote this Lambert series -L(q), but never give it in terms of Thetas or ThetaPrimes. Despite the section heading, they never deliver Sum 1/Fib(n) except in terms of L. It may well be inexpressible without some new special function. (They do give Sum 1/Fib(2*n+1) in Thetas, which is easy.)<rwg But did they give it in terms of q-digamma and I somehow missed or rejected it? http://mathworld.wolfram.com/ReciprocalFibonacciConstant.html claims they do! Could it have been added in a later edition? My (autographed) first edition is unhandy. Here I thought I had capped my 40 yr search, and the answer is sitting in Mathworld. --------------------- Weisstein apparently considers slipping his own results into Mathworld tantamount to public announcement. He did the same thing to NeilB, who worked for months to break the π continued fraction record, unaware that Weisstein had already done so. --rwg

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--rwg

Btw. It's a fun exercise to change between the English and the German WikiPedias (even if you do not understand Krautisch).

Best, jj

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