A203138 Decimal expansion of Gamma(1/24).

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

Offset: 2

Author

N. J. A. Sloane, Dec 29 2011

Keywords

• nonn
• cons

Comments

In the article by Vidunas, the third formula on page 13 is wrong. The exponent of the term K((sqrt(3)-1)/(2*sqrt(2)))^(1/3) is wrong. It should be Gamma(1/24) = Pi^(1/24) * 2^(89/36) * 3^(25/48) * sqrt(1+sqrt(2)) * (sqrt(3)-1)^(1/4) * K(1/sqrt(2))^(1/4) * K((sqrt(3)-1)/(2*sqrt(2)))^(1/6) * K((2-sqrt(3))*(sqrt(3)-sqrt(2)))^(1/2). - Vaclav Kotesovec, Apr 21 2024

Examples

			23.462487693183319881385711469586294930433365134004610164739...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 2..5002
• R. Vidunas, Expressions for values of the Gamma function, arxiv:math/0403510 [math.CA], 2004.
• Index to sequences related to gamma function

Programs

Magma

SetDefaultRealField(RealField(100)); Gamma(1/24); // G. C. Greubel, Mar 10 2018

Maple

evalf(GAMMA(1/24), 110); # Vaclav Kotesovec, Apr 21 2024
evalf(Pi^(1/24) * 2^(89/36) * 3^(25/48) * sqrt(1+sqrt(2)) * (sqrt(3)-1)^(1/4) * EllipticK(1/sqrt(2))^(1/4) * EllipticK((sqrt(3)-1)/(2*sqrt(2)))^(1/6) * EllipticK((2-sqrt(3))*(sqrt(3)-sqrt(2)))^(1/2), 110); # Vaclav Kotesovec, Apr 21 2024

Mathematica

RealDigits[Gamma[1/24], 10, 100][[1]] (* G. C. Greubel, Mar 10 2018 *)

PARI

default(realprecision, 100); gamma(1/24) \\ G. C. Greubel, Mar 10 2018

Formula

From Vaclav Kotesovec, Apr 21 2024: (Start)

Equals 2^(13/12) * 3^(9/16) * Pi^(1/4) * (sqrt(3) - 1)^(1/4) * sqrt((1 + sqrt(2)) * Gamma(1/3) * Gamma(1/4)) * EllipticTheta(3, 0, exp(-Pi*sqrt(6))).

Equals 2^(35/24) * 3^(3/8) * sqrt(Pi*(1 + sqrt(2)) * Gamma(1/12) / (1 + sqrt(3))) * EllipticTheta(3, 0, exp(-Pi*sqrt(6))). (End)