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Reciprocal Lucas Constant

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Closed forms are known for the sums of reciprocals of even-indexed Lucas numbers

P_L^((e))=sum_(n=1)^(infty)1/(L_(2n))
(1)
=sum_(n=1)^(infty)1/(phi^(2n)+phi^(-2n))
(2)
=-1/(4lnphi){pi+i[psi_(phi^2)(1+(ipi)/(4lnphi))-psi_(phi^2)(1-(ipi)/(4lnphi))]}
(3)
=1/4[theta_3^2(phi^(-2))-1]
(4)
=0.566177675...
(5)

(OEIS A153415), where phi is the golden ratio, psi_q^((n))(z) is a q-polygamma function, and theta_n(q) is a Jacobi theta function, and odd-indexed Lucas numbers

P_L^((o))=sum_(n=0)^(infty)1/(L_(2n+1))
(6)
=sum_(n=0)^(infty)(phi^(2n+1))/(phi^(4n+2)-1)
(7)
=L(phi^(-4))-2L(phi^(-2))+L(phi^(-1))
(8)
=1/(4lnphi)[7lnphi-ln(phi^2+1)-4psi_(phi^(-1))(1)+4psi_(phi^(-2))(1)-psi_(phi^(-4))(1)]
(9)
=1/(4lnphi)[psi_(phi^2)(1/2-(ipi)/(2lnphi))-psi_(phi^2)(1/2)+ipi]
(10)
=1.39668...
(11)

(OEIS A153416), where L(beta) is a Lambert series (Borwein and Borwein 1987, pp. 91-92). This gives the reciprocal Lucas constant as

P_L=sum_(n=1)^(infty)1/(L_n)
(12)
=sum_(n=1)^(infty)1/((-phi)^(-n)+phi^n)
(13)
=sum_(n=1)^(infty)(F_n)/(F_(2n))
(14)
=P_L^((e))+P_L^((o))
(15)
=1.96285817...
(16)

(OEIS A093540), where phi is the golden ratio and F_n is a Fibonacci number.

Borwein and Borwein (1987, pp. 94-101) give a number of related beautiful formulas.

See also

Lucas Number, Lambert Series, q-Polygamma Function, Reciprocal Fibonacci Constant

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References

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.Sloane, N. J. A. Sequences A093540, A153415, and A153416 in "The On-Line Encyclopedia of Integer Sequences."

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Reciprocal Lucas Constant

Cite this as:

Weisstein, Eric W. "Reciprocal Lucas Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ReciprocalLucasConstant.html

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