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?
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.