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倒数斐波那契常数

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已知偶数索引斐波那契数的倒数和的闭合形式

P_F^((e))=sum_(n=1)^(infty)1/(F_(2n))
(1)
=sqrt(5)sum_(n=1)^(infty)(phi^(2n))/(phi^(4n)-1)
(2)
=sqrt(5)sum_(n=1)^(infty)(1/(phi^(2n)-1)-1/(phi^(4n)-1))
(3)
=sqrt(5)[L(phi^(-2))-L(phi^(-4))]
(4)
=(sqrt(5))/(8lnphi)[ln5+2psi_(phi^(-4))(1)-4psi_(phi^(-2))(1)]
(5)
=(sqrt(5))/(4lnphi)[psi_(phi^2)(1-(ipi)/(2lnphi))-psi_(phi^2)(1)+ipi]
(6)
=1.5353705...
(7)

(OEIS A153386; Knopp 1990, Ch. 8, Ex. 114; Paszkowski 1997; Horadam 1988; Finch 2003, p. 358; E. Weisstein, Jan. 1, 2009; Arndt 2012), 其中 phi 是黄金比例，psi_q(z)=psi_q^((0))(z) 是一个 q-多伽玛函数，并且 L(beta) 是一个 Lambert 级数 (Borwein 和 Borwein 1987, pp. 91 和 95) 以及奇数索引斐波那契数

P_F^((o))=sum_(n=0)^(infty)1/(F_(2n+1))
(8)
=sqrt(5)sum_(n=0)^(infty)(phi^(2n+1))/(phi^(4n+2)+1)
(9)
=-(sqrt(5))/(4lnphi){pi-i[psi_(phi^2)(1/2-(ipi)/(4lnphi))-psi_(phi^2)(1/2+(ipi)/(4lnphi))]}
(10)
=1/4sqrt(5)theta_2^2(phi^(-2))
(11)
=1/4sqrt(5)[theta_3^2(phi^(-1))-theta_3^2(phi^(-2))]
(12)
=1.824515...
(13)

(OEIS A153387; Landau 1899; Borwein 和 Borwein 1997, p. 94; E. Weisstein, Jan. 1, 2009; Arndt 2012), 其中 theta_n(q) 是一个 Jacobi 椭圆函数。 它们共同给出了倒数斐波那契常数的闭合形式，即

P_F=sum_(n=1)^(infty)1/(F_n)
(14)
=sqrt(5)sum_(n=1)^(infty)1/(phi^(-n)-(-phi)^n)
(15)
=P_F^((e))+P_F^((o))
(16)
=(sqrt(5))/4[(ln5+2psi_(phi^(-4))(1)-4psi_(phi^(-2))(1))/(2lnphi)+theta_2^2(phi^(-2))]
(17)
=3.35988566...
(18)

(OEIS A079586; Horadam 1988; Griffin 1992; Zhao 1999; Finch 2003, p. 358)。 保罗·埃尔德什正式提出了 P_F 的无理性的问题，并且 André-Jeannin (1989) 证明了这个和是无理数。

Borwein 和 Borwein (1987, pp. 94-101) 给出了一些优美的相关公式。

另请参阅

斐波那契数, Lambert 级数, q-多伽玛函数, 倒数卢卡斯常数

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参考文献

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 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. 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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倒数斐波那契常数

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韦斯坦因，埃里克·W. "倒数斐波那契常数。" 来自 MathWorld——Wolfram Web 资源。 https://mathworld.net.cn/ReciprocalFibonacciConstant.html

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