Reciprocal Fibonacci constant
The value of ψ is known to be approximately
No closed formula for ψ is known, but Gosper describes an algorithm for fast numerical approximation of its value. The reciprocal Fibonacci series itself provides O(k) digits of accuracy for k terms of expansion, while Gosper's accelerated series provides O(k2) digits. ψ is known to be irrational; this property was conjectured by Paul Erdős, Ronald Graham, and Leonard Carlitz, and proved in 1989 by Richard André-Jeannin.
The continued fraction representation of the constant is:
- Gosper, William R. (1974), Acceleration of Series, Artificial Intelligence Memo #304, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, pp. p.66.
- André-Jeannin, Richard (1989), "Irrationalité de la somme des inverses de certaines suites récurrentes", C. R. Acad. Sci. Paris Sér. I Math. 308 (19): 539–541, MR 0999451