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

Author

Benoit Cloitre, Mar 10 2002

Keywords

• cons
• easy
• nonn

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

Examples

			3.6256099082219083119306851558676720029951676828800654674333...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 33.
• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, equation 43:4:13 at page 414.

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.

Crossrefs

Cf. A014549, A248557.

Programs

Magma

R:= RealField(100); SetDefaultRealField(R); Gamma(1/4); // G. C. Greubel, Mar 10 2018

Maple

evalf(GAMMA(1/4));

Mathematica

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

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

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)

Gamma(1/4) * 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