TOPICS

Reciprocal Fibonacci Constant

Closed forms are known for the sums of reciprocals of even-indexedFibonaccinumbers

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)
			(7)

(OEISA153386; Knopp 1990, Ch. 8, Ex. 114; Paszkowski 1997; Horadam 1988; Finch 2003, p. 358; E. Weisstein, Jan. 1, 2009; Arndt 2012), where phi is thegolden ratio, psi_q(z)=psi_q^((0))(z) is aq-polygamma function, and L(beta) is aLambert series (Borwein and Borwein 1987, pp. 91 and 95) and odd-indexed Fibonacci numbers

			(8)
			(9)
		$-(sqrt(5))/(4lnphi){pi-i[psi_(phi^2)(1/2-(ipi)/(4lnphi))-psi_(phi^2)(1/2+(ipi)/(4lnphi))]}$	(10)
			(11)
			(12)
			(13)

(OEISA153387; Landau 1899; Borwein and Borwein 1997, p. 94; E. Weisstein, Jan. 1, 2009; Arndt 2012), where theta_n(q) is aJacobi elliptic function. Together, these give a closed form for the reciprocal Fibonacci constant of

			(14)
			(15)
			(16)
			(17)
			(18)

(OEISA079586; Horadam 1988; Griffin 1992; Zhao 1999; Finch 2003, p. 358). The question of the irrationality of P_F 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.

Explore with Wolfram|Alpha

More things to try:

References

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 inPi & 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. France27, 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 sumt^n/(Aalpha^n+Bbeta^n)

."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. SequencesA079586,A153386, andA153387 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.

Referenced on Wolfram|Alpha

Reciprocal Fibonacci Constant

Cite this as:

Weisstein, Eric W. "Reciprocal Fibonacci Constant."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/ReciprocalFibonacciConstant.html

Movatterモバイル変換

Reciprocal Fibonacci Constant

See also

Explore with Wolfram|Alpha

References

Referenced on Wolfram|Alpha

Cite this as:

Subject classifications