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%I #5 Mar 27 2024 21:29:11
%S 4,7,9,6,2,8,8,5,2,3,1,8,8,3,8,5,4,6,3,8,1,0,3,7,0,1,4,0,7,5,1,2,1,5,
%T 8,4,9,8,1,9,5,1,6,3,0,8,0,9,2,3,4,7,7,4,1,8,3,7,3,9,5,7,2,2,0,5,7,8,
%U 3,4,2,6,1,6,7,9,3,5,0,8,9,5,4,9,8,5,7,6,6,1,0,8,0,0,6,2,8,3,1,2,5,4,6,6,6
%N Decimal expansion of Product_{k>=1} (1 + 1/Lucas(k)).
%C Any two of the four constants {A337668, A337669, this, A371526} are algebraically independent over Q, while any three are not (Duverney et al., 2022).
%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>=1} (1 + 1/A000032(k)).
%F Equals 2 * sqrt(5) * A371529.
%F Equals 2 * phi^(1/4) * eta(2*tau_0)^3 * eta(3*tau_0) / (eta(tau_0)^2 * 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 4.79628852318838546381037014075121584981951630809234...
%t With[{eta = DedekindEta, tau0 = Log[GoldenRatio]*I/Pi}, RealDigits[2 * Surd[GoldenRatio, 4] * eta[2*tau0]^3 * eta[3*tau0]/(eta[tau0]^2 * eta[4*tau0]), 10, 120][[1]]]
%o (PARI) prodinf(k = 1, 1 + 1/(fibonacci(k-1) + fibonacci(k+1)))
%Y Cf. A000032, A001622, A002390, A337668, A337669, A371526, A371527, A371528, A371529, A371530.
%K nonn,cons
%O 1,1
%A _Amiram Eldar_, Mar 26 2024
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