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The Reciprocal Fibonacci constant ($\psi$) is defined as

$$\psi=\sum_{k=1}^{\infty} \frac{1}{F_k}$$

where $F_{k}$ is the $k^{th}$ Fibonacci number.

The irrationality of $\psi$ has been proven. Does the Reciprocal Fibonacci constant have a use in mathematics or is is notable simply because it is the value of an interesting sum?

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For my undergrad math research class, I looked at the series $\sum_{k=1}^{\infty} \frac{\epsilon_k}{F_k}$ where $P[\epsilon_k=1]=P[\epsilon_k=-1]=\frac{1}{2}$, to try to find a closed form for the constant, but I didn't get too far... – Nishrito May 15 '12 at 2:25
@Nishrito could you please post your research and send me the link on my Math SE account? – zerosofthezeta Aug 28 '13 at 4:36
up vote 9 down vote accepted

I'd say it's notable because 1) it's a pretty natural thing to write down and ask about, and 2) it has been proved irrational - irrationality proofs are not all that easy to come by, once you have exhausted things like square roots, cube roots, etc. I'm not aware of any place where the number comes up.

EDIT: It is, however, mentioned in these places:

A F Horadam, Elliptic functions and Lambert series in the summation of reciprocals in certain recurrence-generated sequences, Fib Q 26 (1988) 98-114.

P Griffin, Acceleration of the sum of Fibonacci reciprocals, Fib Q 30 (1992) 179-181.

F-Z Zhao, Notes on reciprocal series related to Fibonacci and Lucas numbers, Fib Q 37 (1999) 254-257.

and also in the papers where it, and more general sums of similar type, are proved irrational.

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As far as I know, your constant $\psi$ doesn't arise naturally other than in just studying it from the outset. However, a related constant does make its appearance in a surprising way. That is, $$\beta:=1+\sum_{k=1}^\infty \dfrac{1}{\mathrm F_{2k}},$$or rather its reciprocal $1/\beta\approx0.3944196702$, arises as the solution to a problem in the analysis of bounded real sequences, which is the title question of my paper that addresses it: "How slowly can a bounded sequence cluster?". It is fully answered there and in the follow-up paper "A maximally separated sequence". The two papers are published in Functiones et Approximatio, volumes 46.2 (2012) and 48.1 (2013), which you can read if you have access to a library that subscribes to Project Euclid (the abstracts are freely available on-line). When I set out on the research, I had no inkling that a number with this sort of structure would come up. If this interests you, I can provide more details.

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My university does not have access to that service but I am interested in reading the two papers. Would you consider posting them on the arXiv if the journal permits? – Antonio Vargas May 16 '13 at 6:00
@AntonioVargas: I have assigned copyright of the electronic files to the journal, and it is quite proprietorial about them. But it has given me a generous number of paper offprints that I can distribute freely. I will post copies of the papers to you by airmail. – John Bentin May 16 '13 at 12:48
John, I would appreciate that very much. If you email me at the address listed on my webpage then I will respond with the mailing address for my department. – Antonio Vargas May 16 '13 at 18:42

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