Reciprocal Fibonacci Constant (original) (raw)
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Closed forms are known for the sums of reciprocals of even-indexed Fibonacci numbers
(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
(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
(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.
See also
Fibonacci Number, Lambert Series, _q_-Polygamma Function,Reciprocal Lucas Constant
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References
André-Jeannin, R. "Irrationalité de la somme des inverses de certaines suites récurrentes." C. R. Acad. Sci. Paris 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. https://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.
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Cite this as:
Weisstein, Eric W. "Reciprocal Fibonacci Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ReciprocalFibonacciConstant.html