Use of the Reciprocal Fibonacci constant?

Solution 1:

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.

Solution 2:

As far as I know, the 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.