A090998 Decimal expansion of lim_{k -> +-oo} k^2*(1 - Gamma(1+i/k)) where i^2 = -1 and Gamma is the Gamma function.

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

Offset: 0

Author

Benoit Cloitre, Feb 29 2004

Keywords

• cons
• nonn

Comments

Limit_{k->oo} k*(1-Gamma(1+1/k)) = -Gamma'(1) = gamma = 0.577....

Decimal expansion of the higher-order exponential integral constant gamma(2,1). The higher-order exponential integrals, see A163931, are defined by E(x,m,n) = x^(n-1)*Integral_{t=x..oo} (E(t,m-1,n)/t^n) dt for m >= 1 and n >= 1, with E(x,m=0,n) = exp(-x). The series expansions of the higher-order exponential integrals are dominated by the gamma(k,n) and the alpha(k,n) constants, see A163927. - Johannes W. Meijer and Nico Baken, Aug 13 2009

Examples

			G(2,1) = 0.9890559953279725553953956515...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No. 3, April 1987, pp. 209-211.

Crossrefs

Cf. A163931 (E(x,m,n)), A163927 (alpha(k,n)), A001620 (gamma).

The structure of the G(k,n=1) formulas lead (replace gamma with G and Zeta with Z) to A036039. - Johannes W. Meijer and Nico Baken, Aug 13 2009

Cf. A081855.

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); (6*EulerGamma(R)^2 + Pi(R)^2)/12; // G. C. Greubel, Feb 01 2019

Maple

ncol:=1; nmax:=5; kmax:=nmax; for n from 1 to nmax do G(0,n):=1 od: for n from 1 to nmax do for k from 1 to kmax do G(k,n):= expand((1/k)*((gamma-sum(p^(-1),p=1..n-1))* G(k-1,n)+sum((Zeta(k-i)-sum(p^(-(k-i)),p=1..n-1))*G(i,n),i=0..k-2))) od; od: for k from 0 to kmax do G(k,ncol):=G(k,ncol) od; # Johannes W. Meijer and Nico Baken, Aug 13 2009

Mathematica

RealDigits[(6*EulerGamma^2 + Pi^2)/12, 10, 104][[1]] (* Jean-François Alcover, Mar 04 2013 *)

PARI

default(realprecision, 100); (6*Euler^2 +Pi^2)/12 \\ G. C. Greubel, Feb 01 2019

Sage

numerical_approx((6*euler_gamma^2 + pi^2)/12, digits=100) # G. C. Greubel, Feb 01 2019

Formula

From Johannes W. Meijer and Nico Baken, Aug 13 2009: (Start)

G(2,1) = gamma(2,1) = gamma^2/2 + Pi^2/12.

G(k,n) = (1/k)*(gamma*G(k-1,n)) - (1/k)*Sum_{p=1..n-1} (p^(-1))* G(k-1,n) + (1/k) * Sum_{i=0..k-2} (Zeta(k-i) * G(i,n)) - (1/k)*Sum_{i=0..k-2}(Sum_{p=1..n-1} (p^(i-k)) * G(i,n)) with G(0,n) = 1 for k >= 0 and n >= 1.

G(k,n+1) = G(k,n) - G(k-1,n)/n.

GF(z,n) = GAMMA(n-z)/GAMMA(n).

(gamma - G(1,n)) = A001008(n-1)/A002805(n-1) for n >= 2. (End)

Equals A081855/2. - Hugo Pfoertner, Mar 12 2024