A327995 Decimal expansion of Gamma(3/4)/Pi^(1/4).

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

Offset: 0

Author

Peter Luschny, Oct 24 2019

Keywords

• nonn
• cons

Comments

The function df(x) = 2^(x/2)*(2/Pi)^(sin(Pi*x/2)^2/2)*Gamma(x/2+1) interpolates the double factorials A006882 and extends them analytically. df(-1/2) is the given constant. Extending also the notation this can be written as (-1/2)!! = (-1/4)!/Pi^(1/4).

Examples

			0.92044178783559098393491713074999098122949892...
		

Links

• Jean-Paul Allouche, Samin Riasat, and Jeffrey Shallit, More infinite products: Thue-Morse and the Gamma function, The Ramanujan Journal, Vol. 49 (2019), pp. 115-128; arXiv preprint, arXiv:1709.03398 [math.NT], 2017.

Crossrefs

Cf. A010060, A006882, A327996.

Programs

Maple

Digits := 100: GAMMA(3/4)/Pi^(1/4)*10^86:
ListTools:-Reverse(convert(floor(%), base, 10));

Mathematica

RealDigits[Gamma[3/4]/Surd[Pi,4],10,120][[1]] (* Harvey P. Dale, Jun 13 2020 *)

PARI

gamma(3/4)/Pi^(1/4) \\ Michel Marcus, Oct 24 2019

Formula

Equals (-1/4)!/Pi^(1/4).

From Amiram Eldar, Feb 04 2024: (Start)

Equals Pi^(3/4)*sqrt(2)/Gamma(1/4).

Equals Product_{k>=0} ((4*k+1)*(4*k+4)/((4*k+2)*(4*k+3)))^A010060(k) (Allouche et al., 2019). (End)