(OEIS A153386; Horadam 1988; Knopp 1990, Ch. 8, Ex. 114; Paszkowski 1997; Finch 2003, p. 358; E. Weisstein, Jan. 1,
2009; Arndt 2012), where is the golden ratio,
is a q-polygamma function, and
for
is a Lambert series (Borwein and Borwein 1987,
pp. 91 and 95).
(OEIS A153387; Landau 1899; Borwein and Borwein 1987, p. 94; E. Weisstein, Jan. 1, 2009; Arndt 2012), where is a Jacobi
theta function. Carlitz (1967) gives the identity
Together, the even- and odd-indexed sums give a closed
form for the reciprocal Fibonacci constant of
(15)
(16)
(17)
(18)
(19)
(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. Paris308, 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.Carlitz, L. "An Infinite Series
Equality." Solution to Problem B-110. Fib. Quart.5, 469-470,
1967. https://www.fq.math.ca/Scanned/5-5/elementary5-5.pdf.Duverney,
D. "Irrationalité de la somme des inverses de la suite de Fibonacci."
Elem. Math.52, 31-36, 1997. https://doi.org/10.1007/S000170050008.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 inverses des nombres de Fibonacci." Bull.
Soc. Math. France27, 298-300, 1899. https://www.numdam.org/article/BSMF_1899__27__297_0.pdf.Michon,
G. P. "Final Answers: Numerical Constants." https://www.numericana.com/answer/constants.htm#prevost.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.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.
for 
.
![sqrt(5)[L(phi^(-2))-L(phi^(-4))]](/images/equations/ReciprocalFibonacciConstant/Inline14.svg)
![(sqrt(5))/(8lnphi)[ln5+2psi_(phi^(-4))(1)-4psi_(phi^(-2))(1)]](/images/equations/ReciprocalFibonacciConstant/Inline17.svg)
![(sqrt(5))/(4lnphi)[psi_(phi^2)(1-(ipi)/(2lnphi))-psi_(phi^2)(1)+ipi]](/images/equations/ReciprocalFibonacciConstant/Inline20.svg)
is the golden ratio, 
is a q-polygamma function, and 
for 
is a Lambert series (Borwein and Borwein 1987,
pp. 91 and 95).
![-(sqrt(5))/(4lnphi){pi-i[psi_(phi^2)(1/2-(ipi)/(4lnphi))-psi_(phi^2)(1/2+(ipi)/(4lnphi))]}](/images/equations/ReciprocalFibonacciConstant/Inline36.svg)
![1/4sqrt(5)[theta_3^2(phi^(-1))-theta_3^2(phi^(-2))]](/images/equations/ReciprocalFibonacciConstant/Inline42.svg)
is a Jacobi
theta function. Carlitz (1967) gives the identity
is a Lucas number.
![(sqrt(5))/4[(ln5+2psi_(phi^(-4))(1)-4psi_(phi^(-2))(1))/(2lnphi)+theta_2^2(phi^(-2))]](/images/equations/ReciprocalFibonacciConstant/Inline59.svg)
was formally raised by Paul Erdős and this sum was proved
to be irrational by André-Jeannin (1989).
." J. Number Th. 73,
139-161, 1998.Sloane, N. J. A. Sequences