A068466 Decimal expansion of Gamma(1/4).

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

Offset

1,1

Comments

Nesterenko proves that this constant is transcendental (he cites Chudnovsky as the first to show this); in fact it is algebraically independent of Pi and e^Pi over Q. - Charles R Greathouse IV, Nov 11 2013

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 33.

Links

Harry J. Smith, Table of n, a(n) for n = 1..1000

William Duke and Özlem Imamoḡlu, Special values of multiple gamma functions, Journal de théorie des nombres de Bordeaux, Vol. 18. No. 1 (2006), pp. 113-123.

Greg J. Fee and Simon Plouffe, Gamma(1/4) to 25000 digits.

Yu. V. Nesterenko, Modular functions and transcendence questions, Sbornik: Mathematics, Vol. 187, No. 9 (1996), pp. 1319-1348. (English translation)

Simon Plouffe, Gamma(1/4) to 250000 digits.

Dan Romik, On Viazovska's modular form inequalities, arXiv:2303.13427 [math.NT], 2023.

Eric Weisstein's World of Mathematics, Gamma Function.

Wikipedia, Particular values of the Gamma function: General rational arguments.

Index to sequences related to the Gamma function.

Index entries for transcendental numbers.

Formula

From Amiram Eldar, Jun 12 2021: (Start)

Equals sqrt(2*sqrt(2*Pi^3)*G), where G is Gauss's constant (A014549).

Equals (2*Pi)^(3/4) * Product_{k>=1} tanh(k*Pi/2) (Duke and Imamoḡlu, 2006). (End)

This * A068465 = A063448. - R. J. Mathar, May 22 2024

Equals Product_{n>=1} exp((2*(6*n + 1)*(1 - beta(n)) - (eta(n) - 1))/(4*n)), where eta(n) and beta(n) are the Dirichlet eta and beta functions, respectively. - Antonio Graciá Llorente, Sep 05 2024

Example

3.6256099082219083119306851558676720029951676828800654674333...

Maple

evalf(GAMMA(1/4));

Mathematica

RealDigits[Gamma[1/4], 10, 110][[1]] (* Bruno Berselli, Dec 13 2012 *)

Prog

(PARI) default(realprecision, 1080); x=gamma(1/4); for (n=1, 1000, d=floor(x); x=(x-d)*10; write("b068466.txt", n, " ", d)); \\ Harry J. Smith, Apr 19 2009
(Magma) R:= RealField(100); SetDefaultRealField(R); Gamma(1/4); // G. C. Greubel, Mar 10 2018

Crossrefs

Cf. A014549, A248557.

Sequence in context: A319513 A160590 A111952 * A194076 A194040 A194065

Adjacent sequences: A068463 A068464 A068465 * A068467 A068468 A068469

Keyword

cons,easy,nonn

Author

Benoit Cloitre, Mar 10 2002

Status

approved