A268895 Decimal expansion of the upper bound of 1/Gamma(x) - x on the unit interval x = [0,1].

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

Offset

-1,1

Comments

Gamma(x) stands for the gamma function (Euler's integral of the second kind).

On the unit interval the function 1/Gamma(x) may be bounded from below and from above as follows: x <= 1/Gamma(x) <= x + C, where C = 0.072186279... is the constant which we introduced above. Numerical simulations show that these lower and upper bounds are both quite accurate. Some other bounds for 1/Gamma(x) may be found in the reference given below.

Numerically, the value of C is quite close to the first Stieltjes constant with the opposite sign (see A082633).

Links

Table of n, a(n) for n=-1..102.

Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi, arXiv:1408.3902 [math.NT], 2014-2016.

Formula

Equals 1/Gamma(x_0) - x_0, where x_0 is the unique positive root of the equation Gamma(x) + Psi(x) = 0 (see A268893).

Example

0.0721862796822481164934370114884600281187017754898161...

Maple

Digits:= 500; x0:=fsolve(Psi(x)+GAMMA(x)=0, x): evalf(1/GAMMA(x0)-x0, 120);

Mathematica

y = FindRoot[Gamma[x]+PolyGamma[x]==0, {x, 0.6}, WorkingPrecision->120][[1, 2]]; N[1/Gamma[y] - y, 120] // RealDigits[#, 10, 104] & // First

Prog

(PARI) default(realprecision, 500); x0=solve(x = 0.60, 0.68, gamma(x)+psi(x)); 1/gamma(x0)-x0

Crossrefs

Cf. A268893, A268911.

Sequence in context: A060991 A120455 A377813 * A108433 A274570 A334056

Adjacent sequences: A268892 A268893 A268894 * A268896 A268897 A268898

Keyword

nonn,cons

Author

Iaroslav V. Blagouchine, Feb 15 2016

Status

approved