A248557 -id:A248557 - cp's OEIS frontend

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

Benoit Cloitre, Mar 10 2002

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

			3.6256099082219083119306851558676720029951676828800654674333...

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.

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.

Cf. A014549, A248557.

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

Maple
```
evalf(GAMMA(1/4));
```

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

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

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

A014549 Decimal expansion of 1 / M(1,sqrt(2)) (Gauss's constant).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, N. J. A. Sloane

On May 30, 1799, Gauss discovered that this number is also equal to (2/Pi)*Integral_{t=0..1} 1/sqrt(1-t^4).

M(a,b) is the limit of the arithmetic-geometric mean iteration applied repeatedly starting with a and b: a_0 = a, b_0 = b, a_{n+1} = (a_n + b_n)/2, b_{n+1} = sqrt(a_n*b_n).

			0.8346268416740731862814297327990468...

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, page 5.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.5.4 and 6.1, pp. 34, 420.
J. R. Goldman, The Queen of Mathematics, 1998, p. 92.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Markus Faulhuber, An application of hypergeometric functions to heat kernels on rectangular and hexagonal tori and a "Weltkonstante"-or-how Ramanujan split temperatures, The Ramanujan Journal volume 54, pages 1-27 (2021). See pp. 4 and 24.
Markus Faulhuber, Anupam Gumber, and Irina Shafkulovska, The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators, arXiv:2209.04202 [math.CA], 2022, p. 2.
Alessandro Languasco and Pieter Moree, Euler constants from primes in arithmetic progression, arXiv:2406.16547 [math.NT], 2024. See p. 10.
Eric Weisstein's World of Mathematics, Gauss's Constant.
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean.
Index entries for transcendental numbers.

Cf. A053002, A053003, A053004, A062539, A093341, A244644, A248557.

SetDefaultRealField(RealField(100)); R:= RealField(); Sqrt(Pi(R)/2)/Gamma(3/4)^2; // G. C. Greubel, Aug 17 2018

evalf(1/GaussAGM(1, sqrt(2)), 144);  # Alois P. Heinz, Jul 05 2023

RealDigits[Gamma[1/4]^2/(2*Pi^(3/2)*Sqrt[2]), 10, 105][[1]] (* or: *)
RealDigits[1/ArithmeticGeometricMean[1, Sqrt[2]], 10, 105][[1]] (* Jean-François Alcover, Dec 13 2011, updated Nov 11 2016, after Eric W. Weisstein *)
First[RealDigits[N[EllipticTheta[4, Exp[-Pi]]^2, 90]]] (* Stefano Spezia, Sep 29 2022 *)

default(realprecision, 20080); x=10*agm(1, sqrt(2))^-1; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b014549.txt", n, " ", d)); \\ Harry J. Smith, Apr 20 2009

1/agm(sqrt(2),1) \\ Charles R Greathouse IV, Feb 04 2015

sqrt(Pi/2)/gamma(3/4)^2 \\ Charles R Greathouse IV, Feb 04 2015

from mpmath import mp, agm, sqrt
mp.dps=105
print([int(z) for z in list(str(1/agm(sqrt(2)))[2:-1])]) # Indranil Ghosh, Jul 11 2017

Equals (lim_{k->oo} p(k))/(1+i) and (lim_{k->oo} q(k))/(1+i), where i is the imaginary unit, p(0) = 1, q(0) = i, p(k+1) = 2*p(k)*q(k)/(p(k)+q(k)) and q(k+1) = sqrt(p(k)*q(k)) for k >= 0. - A.H.M. Smeets, Jul 26 2018
Equals the infinite quotient product (3/4)*(6/5)*(7/8)*(10/9)*(11/12)*(14/13)*(15/16)*... . - James Maclachlan, Jul 28 2019
Equals (9/15)*hypergeom([1/2, 3/4], [9/4], 1). - Peter Bala, Mar 03 2022
Equals A062539 / Pi. - Amiram Eldar, May 04 2022
From Stefano Spezia, Sep 29 2022: (Start)
Equals theta4(exp(-Pi))^2.
Equals sqrt(2)*A093341/Pi. (End)
Equals Sum_{k>=0} (-1)^k * binomial(2*k,k)^2/16^k. - Amiram Eldar, Jul 04 2023
From Gerry Martens, Jul 31 2023: (Start)
Equals 2*Gamma(5/4)/(sqrt(Pi)*Gamma(3/4)).
Equals hypergeom([1/4, -2/4], [1], 1). (End)
Equals A248557^2. - Hugo Pfoertner, Jun 28 2024

Extended to 105 terms by Jean-François Alcover, Dec 13 2011

a(104) corrected by Andrew Howroyd, Feb 23 2018

A251992 Decimal expansion of the double infinite sum (negated) sum_{m=1..infinity} sum_{k=0..infinity} (-1)^m/((2k+1)^2+m^2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Dec 12 2014

			-0.480751144424109780520862631352408574248444731679469...

MathOverflow, How to calculate the infinite sum of this double series?

Cf. A175573.

RealDigits[-Pi*(Pi-Log[2])/16, 10, 103] // First

-Pi*(Pi-log(2))/16.
Also equals sum_{m=1..infinity} (-1)^m*Pi*tanh(m*Pi/2)/(4*m).
Also equals -Pi^2/16 - (Pi/4)*log(theta_2(0,exp(-Pi))) + (Pi/4)*log(theta_3(0,exp(-Pi))), where 'theta' is the elliptic theta function, that is -Pi^2/16 - (Pi/4)*log(A248557) + (Pi/4)*log(A175573).

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A068466 Decimal expansion of Gamma(1/4).

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A014549 Decimal expansion of 1 / M(1,sqrt(2)) (Gauss's constant).

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A251992 Decimal expansion of the double infinite sum (negated) sum_{m=1..infinity} sum_{k=0..infinity} (-1)^m/((2k+1)^2+m^2).

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