Reciprocal Fibonacci Constant
Closed forms are known for the sums of reciprocals of even-indexed Fibonacci numbers
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(1)
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(2)
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(3)
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 |  | ![sqrt(5)[L(phi^(-2))-L(phi^(-4))]](/images/equations/ReciprocalFibonacciConstant/Inline12.gif) |
(4)
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 |  | ![(sqrt(5))/(8lnphi)[ln5+2psi_(phi^(-4))(1)-4psi_(phi^(-2))(1)]](/images/equations/ReciprocalFibonacciConstant/Inline15.gif) |
(5)
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 |  | ![(sqrt(5))/(4lnphi)[psi_(phi^2)(1-(ipi)/(2lnphi))-psi_(phi^2)(1)+ipi]](/images/equations/ReciprocalFibonacciConstant/Inline18.gif) |
(6)
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(7)
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(OEIS A153386; Knopp 1990, Ch. 8, Ex. 114; Paszkowski 1997; Horadam 1988; Finch 2003, p. 358; E. Weisstein, Jan. 1,
2009; Arndt 2012), where 
is the golden
ratio, 
is a q-polygamma function, and 
is a Lambert
series (Borwein and Borwein 1987, pp. 91 and 95) and odd-indexed Fibonacci
numbers
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(8)
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(9)
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 |  | ![-(sqrt(5))/(4lnphi){pi-i[psi_(phi^2)(1/2-(ipi)/(4lnphi))-psi_(phi^2)(1/2+(ipi)/(4lnphi))]}](/images/equations/ReciprocalFibonacciConstant/Inline33.gif) |
(10)
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(11)
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 |  | ![1/4sqrt(5)[theta_3^2(phi^(-1))-theta_3^2(phi^(-2))]](/images/equations/ReciprocalFibonacciConstant/Inline39.gif) |
(12)
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(13)
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(OEIS A153387; Landau 1899; Borwein and Borwein 1997, p. 94; E. Weisstein, Jan. 1, 2009; Arndt 2012), where 
is a
Jacobi elliptic function. Together, these
give a closed form for the reciprocal Fibonacci constant of
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(14)
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(15)
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(16)
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 |  | ![(sqrt(5))/4[(ln5+2psi_(phi^(-4))(1)-4psi_(phi^(-2))(1))/(2lnphi)+theta_2^2(phi^(-2))]](/images/equations/ReciprocalFibonacciConstant/Inline55.gif) |
(17)
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(18)
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(OEIS A079586; Horadam 1988; Griffin 1992; Zhao 1999; Finch 2003, p. 358). The question of the irrationality of 
was formally
raised by Paul Erdős and this sum was proved to be irrational by André-Jeannin
(1989).
Borwein and Borwein (1987, pp. 94-101) give a number of beautiful related formulas.
André-Jeannin, R. "Irrationalité de la somme des inverses de certaines suites récurrentes." C. R. Acad. Sci. Paris Sér. I Math. 308, 539-541, 1989.
Apéry, F. "Roger Apéry, 1916-1994: A Radical Mathematician." Math. Intell. 18, No. 2, 54-61, 1996.
Arndt, J. "On Computing the Generalized Lambert Series." 24 Jun 2012. http://arxiv.org/abs/1202.6525.
Borwein, J. M. and Borwein, P. B. "Evaluation of Sums of Reciprocals of Fibonacci Sequences." §3.7 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 91-101, 1987.
Bundschuh, P. and Väänänen, K. "Arithmetical Investigations of a Certain Infinite Product." Compos. Math. 91, 175-199, 1994.
Duverney, D. "Irrationalité de la somme des inverses de la suite de Fibonacci." Elem. Math. 52, 31-36, 1997.
Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 358, 2003.
Griffin, P. "Acceleration of the Sum of Fibonacci Reciprocals." Fib. Quart. 30, 179-181, 1992.
Horadam, A. F. "Elliptic Functions and Lambert Series in the Summation of Reciprocals in Certain Recurrence-Generated Sequences." Fib. Quart. 26, 98-114, 1988.
Knopp, K. Theory and Application of Infinite Series. New York: Dover, 1990.
Landau, E. "Sur la série des inverse de nombres de Fibonacci." Bull. Soc. Math. France 27, 298-300, 1899.
Paszkowski, S. "Fast Convergent Quasipower Series for Some Elementary and Special Functions." Comput. Math. Appl. 33, 181-191, 1997.
Prévost, M. "On the Irrationality of 
."
J. Number Th. 73, 139-161, 1998.
Michon, G. P. "Final Answers: Numerical Constants." http://home.att.net/~numericana/answer/constants.htm#prevost.
Sloane, N. J. A. Sequences A079586, A153386, and A153387 in "The On-Line Encyclopedia of Integer Sequences."
Zhao, F.-Z. "Notes of Reciprocal Series Related to the Fibonacci and Lucas Numbers." Fib. Quart. 37, 254-257, 1999.
Weisstein, Eric W. "Reciprocal Fibonacci Constant." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ReciprocalFibonacciConstant.html
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