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Decimal expansion of Product_{k>=1} (1 - (-1)^k/Lucas(k)).
5

%I #4 Mar 27 2024 21:30:21

%S 1,5,0,2,1,8,0,0,4,4,3,3,2,4,5,6,7,6,9,1,2,0,7,6,2,5,8,1,7,6,5,5,6,9,

%T 9,8,8,0,2,7,1,5,2,5,8,0,8,8,8,8,8,3,6,4,4,5,1,5,0,1,5,5,1,1,7,0,7,8,

%U 7,4,1,9,3,3,3,7,5,9,4,6,3,2,9,9,3,4,4,3,7,1,9,2,1,5,9,4,8,3,9,2,4,1,0,8,8

%N Decimal expansion of Product_{k>=1} (1 - (-1)^k/Lucas(k)).

%H Daniel Duverney, Carsten Elsner, Masanobu Kaneko, and Yohei Tachiya, <a href="https://doi.org/10.1007/s40993-022-00328-7">A criterion of algebraic independence of values of modular functions and an application to infinite products involving Fibonacci and Lucas numbers</a>, Research in Number Theory, Vol. 8 (2022), Article 31; <a href="https://www2.math.kyushu-u.ac.jp/~mkaneko/papers/DEKT.pdf">alternative link</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DedekindEtaFunction.html">Dedekind Eta Function</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dedekind_eta_function">Dedekind eta function</a>.

%F Equals Product_{k>=2} (1 - (-1)^k/A000032(k)).

%F Equals (2*sqrt(5)) * A371526.

%F Equals 2 * phi^(1/4) * eta(2*tau_0)^2 * eta(6*tau_0) / (eta(3*tau_0) * eta(4*tau_0)), where phi is the golden ratio (A001622), tau_0 = i*log(phi)/Pi, and i = sqrt(-1) (Duverney et al., 2022).

%e 1.50218004433245676912076258176556998802715258088888...

%t With[{eta = DedekindEta, tau0 = Log[GoldenRatio]*I/Pi}, RealDigits[2 * Surd[GoldenRatio, 4] * eta[2*tau0]^2 * eta[6*tau0]/(eta[3*tau0] * eta[4*tau0]), 10, 120][[1]]]

%o (PARI) prodinf(k = 1, 1 - (-1)^k/(fibonacci(k-1) + fibonacci(k+1)))

%Y Cf. A000032, A001622, A002390, A337668, A337669, A371525, A371526, A371527, A371528, A371529.

%K nonn,cons

%O 1,2

%A _Amiram Eldar_, Mar 26 2024