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Decimal expansion of (PolyGamma(1,(1+sqrt(5))/4)-PolyGamma(1,(3+sqrt(5))/4))/2.
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%I #26 Jan 01 2025 14:42:14

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

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

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

%N Decimal expansion of (PolyGamma(1,(1+sqrt(5))/4)-PolyGamma(1,(3+sqrt(5))/4))/2.

%D Bruce C. Berndt and Robert A. Rankin, Ramanujan: Letters and Commentary, Amer. Math. Soc., 1995, p. 57.

%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 424.

%D G. H. Hardy, Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, Cambridge, 1940, p. 8, eq. (1.9).

%H Bruce C. Berndt and Atul Dixit, <a href="https://doi.org/10.46298/hrj.2021.7429">Ramanujan's Beautiful Integrals</a>, Hardy-Ramanujan Journal, Vol. 43 (2020), pp. 69-82. See Theorem 3.2, p. 72.

%H G. H. Hardy, <a href="https://doi.org/10.1112/plms/s2-19.1.1-v">Srinivasa Ramanujan</a>, obituary notice, Proceedings of the London Mathematical Society, Vol. s2-19, No. 1 (1921), pp. xl-xlix. See p. xlix, eq. (2).

%H Oskar Perron, <a href="https://publikationen.badw.de/de/003383728">Über die Preeceschen Kettenbrüche</a>, Sitz. Bayer. Akad. Wiss. München Math. Phys. Kl. (1953), pp. 21-56.

%H C. T. Preece, <a href="https://doi.org/10.1112/jlms/s1-6.1.22">Theorems Stated by Ramanujan (X)</a>, J. London Math. Soc., Vol. s1-6, No. 1 (1931), pp. 22-32.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanContinuedFractions.html">Ramanujan Continued Fractions</a>.

%F From _Amiram Eldar_, Jan 01 2025: (Start)

%F Equals 4 * Integral_{x>=0} x * exp(-sqrt(5)*x) * sech(x) dx.

%F Equals 1/(1 + 1^2/(1 + 1^2/(1 + 2^2/(1 + 2^2/(1 + 3^2/(1 + 3^2/(1 + 4^2/(1 + 4^2/(1 + ... ))))))))). (End)

%e 0.56830000314623517876033041103317515140752667478254...

%t 4*Integrate[(x*Sech[x])/E^(Sqrt[5]*x), {x, 0, Infinity}]

%t RealDigits[(PolyGamma[1,(1+Sqrt[5])/4]-PolyGamma[1,(3+Sqrt[5])/4])/2, 10, 100][[1]] (* _Vaclav Kotesovec_, Aug 16 2015 *)

%o (PARI)

%o polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));

%o (polygamma(1, (1+sqrt(5))/4) - polygamma(1, (3+sqrt(5))/4))/2 \\ _Gheorghe Coserea_, Sep 30 2018

%Y Cf. A091660.

%K nonn,cons

%O 0,1

%A _Eric W. Weisstein_, Jan 26 2004