A091670 Decimal expansion of Gamma(1/4)^4/(4*Pi^3).

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

Offset

1,2

Comments

Watson's first triple integral.

This is also the value of F. Morley's series from 1902 Sum_{k=0..n} (risefac(k,1/2)/k!)^3 = hypergeometric([1/2,1/2,1/2],[1,1],1) with the rising factorial risefac(n,x). See A277232, also for the Hardy reference and a MathWorld link. - Wolfdieter Lang, Nov 11 2016

This constant is transcendental due to a result of Nesterenko, who proves that Gamma(1/4) is algebraically independent of Pi. - Charles R Greathouse IV, Aug 19 2025

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.9, p. 324.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 256, 6.1.17, p. 557, 15.1.26.

M. L. Glasser and I. J. Zucker, Extended Watson integrals for the cubic lattices, Proc. Nat. Acad. Sci., Vol. 74, No. 5 (1977), p. 1800-1801.

M. B. McNeil, Problem 612, Mathematics Magazine Vol. 39, No. 1 (1966), p. 70; A Triple Integral, Solution to Problem 612, by Henry T. Fettis, ibid., Vol. 39, No. 4 (1966), pp. 251-253; Comment on Problem 612, by William D. Fryer, ibid., Vol. 40, No. 1 (1967), pp. 52-53.

Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (6.5.1)

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

Tito Piezas III, Watson's triple integrals.

William Squire, Problem 676, Mathematics Magazine, Vol. 40, No. 5 (1967), p. 279; A Factorial Quotient, Solution to Problem 676, by H. W. Gould, ibid., Vol. 41, No. 3 (1968), pp. 165-166.

Eric Weisstein's World of Mathematics, Watson's Triple Integrals.

I. J. Zucker, 70+years of the Watson integrals, J. Stat. Phys., Vol. 145, No. 3 (2011), pp. 591-612.

Index entries for transcendental numbers.

Formula

From Joerg Arndt, Nov 27 2010: (Start)

Equals 1/agm(1,sqrt(1/2))^2.

Equals Gamma(1/4)^4 / (4*Pi^3) = Pi / (Gamma(3/4))^4 = hypergeom([1/2,1/2],[1],1/2)^2, see the two Abramowitz - Stegun references. (End)

Equals the square of A175574. Equals A000796/A068465^4. - R. J. Mathar, Jun 17 2016

Equals hypergeom([1/2,1/2,1/2],[1,1],1) - Wolfdieter Lang, Nov 12 2016

Equals Sum_{k>=0} binomial(2*k,k)^3/2^(6*k). - Amiram Eldar, Aug 26 2020

Equals Sum_{k>=0} A000118(k) / exp(k*Pi). - Simon Plouffe, Sep 09 2025

Equals A249189 / Pi. - Amiram Eldar, Apr 10 2026

Example

1.39320392968567685918424626032536824265748121751561787897...

Maple

Pi/GAMMA(3/4)^4 ; evalf(%) ; # R. J. Mathar, Jun 17 2016

Mathematica

RealDigits[ N[ Gamma[1/4]^4/(4*Pi^3), 102]][[1]] (* Jean-François Alcover, Nov 12 2012, after Eric W. Weisstein *)

Prog

(PARI) 1/agm(sqrt(1/2), 1)^2 \\ Charles R Greathouse IV, Mar 03 2016
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Gamma(1/4)^4/(4*Pi(R)^3); // G. C. Greubel, Oct 26 2018

Crossrefs

Cf. A000118, A091671, A091672, A249189, A277232, A293238 (inverse), A068466.

Sequence in context: A336501 A016674 A264918 * A388803 A375503 A201416

Adjacent sequences: A091667 A091668 A091669 * A091671 A091672 A091673

Keyword

nonn,cons

Author

Eric W. Weisstein, Jan 27 2004

Status

approved