Reciprocal Fibonacci constant

The reciprocal Fibonacci constant {{mvar|ψ}} is the sum of the reciprocals of the Fibonacci numbers:

$\psi = \sum_{k=1}^{\infty} \frac{1}{F_k} = \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{8} + \frac{1}{13} + \frac{1}{21} + \cdots.$

Because the ratio of successive terms tends to the reciprocal of the golden ratio, which is less than 1, the ratio test shows that the sum converges.

The value of {{mvar|ψ}} is approximately

$\psi = 3.359885666243177553172011302918927179688905133732\dots$ {{OEIS|A079586}}.

With {{mvar|k}} terms, the series gives {{math|O(k)}} digits of accuracy. Bill Gosper derived an accelerated series which provides {{math|O(k²)}} digits.{{citation

| last = Gosper

| first = William R.

| authorlink = Bill Gosper

| year = 1974

| title = Acceleration of Series

| page = 66

| url = http://dspace.mit.edu/handle/1721.1/6088

| publisher = Artificial Intelligence Memo #304, Artificial Intelligence Laboratory, Massachusetts Institute of Technology| hdl = 1721.1/6088

}}.

{{mvar|ψ}} is irrational, as was conjectured by Paul Erdős, Ronald Graham, and Leonard Carlitz, and proved in 1989 by Richard André-Jeannin.{{citation

| last = André-Jeannin

| first = Richard

| title = Irrationalité de la somme des inverses de certaines suites récurrentes

| url = http://gallica.bnf.fr/ark:/12148/bpt6k5686125p/f9.image

| journal = Comptes Rendus de l'Académie des Sciences, Série I

| volume = 308

| year = 1989

| issue = 19

| pages = 539–541

|mr=0999451}}

Its simple continued fraction representation is:

$\psi = [3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,2,4,8,6,30,50,1,6,3,3,2,7,2,3,1,3,2, \dots] \!\,$ {{OEIS|A079587}}.

Generalization and related constants

In analogy to the Riemann zeta function, define the Fibonacci zeta function as

$\zeta_F(s) = \sum_{n=1}^\infty \frac{1}{(F_n)^s} = \frac{1}{1^s} + \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{5^s} + \frac{1}{8^s} + \cdots$ for complex number {{mvar|s}} with {{math|Re(s) > 0}}, and its analytic continuation elsewhere. Particularly the given function equals {{mvar|ψ}} when {{math|1=s = 1}}.{{citation

| last = Murty | first = M. Ram

| editor1-last = Prasad | editor1-first = D.

| editor2-last = Rajan | editor2-first = C. S.

| editor3-last = Sankaranarayanan | editor3-first = A.

| editor4-last = Sengupta | editor4-first = J.

| contribution = The Fibonacci zeta function

| isbn = 978-93-80250-49-6

| mr = 3156859

| pages = 409–425

| publisher = Tata Institute of Fundamental Research

| series = Tata Institute of Fundamental Research Studies in Mathematics

| title = Automorphic representations and {{mvar|L}}-functions

| volume = 22

| year = 2013}}

It was shown that:

The value of {{math|ζ{{sub|F }}(2s)}} is transcendental for any positive integer {{mvar|s}}, which is similar to the case of even-index Riemann zeta-constants {{math|ζ(2s)}}.{{Cite web |last=Waldschmidt |first=Michel|author-link=Michel Waldschmidt |date=January 2022 |title=Transcendental Number Theory: recent results and open problems. |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TNTOpenPbs|type=Lecture slides}}
The constants {{math|ζ{{sub|F }}(2)}}, {{math|ζ{{sub|F }}(4)}} and {{math|ζ{{sub|F }}(6)}} are algebraically independent.
Except for {{math|ζ{{sub|F }}(1)}} which was proved to be irrational, the number-theoretic properties of {{math|ζ{{sub|F }}(2s + 1)}} (whenever s is a non-negative integer) are mostly unknown.

References

External links

{{MathWorld|title = Reciprocal Fibonacci Constant | urlname = ReciprocalFibonacciConstant}}

Category:Mathematical constants

Category:Fibonacci numbers

Category:Irrational numbers

Reciprocal Fibonacci constant

Generalization and related constants

See also

References

External links