A163930 Duplicate of A090998.

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

Johannes W. Meijer and Nico Baken, Aug 13 2009, Aug 17 2009

Keywords

• dead

Comments

The higher order exponential integrals, see A163931, are defined by E(x,m,n) = x^(n-1)*int(E(t,m-1,n)/t^n, t=x..infinity) 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.

The values of the gamma(k,n) = G(k,n) coefficients can be determined with the Maple program.

Examples

			G(2,1) = 0.9890559953279725553953956515...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• 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)).

G(1,1) equals A001620 (gamma).

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

The structure of the G(k,n=1) formulas lead (replace gamma by G and Zeta by Z) to A036039.

Programs

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;

Mathematica

RealDigits[ N[ EulerGamma^2/2 + Pi^2/12, 105]][[1]] (* Jean-François Alcover, Nov 07 2012, from 1st formula *)

Formula

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).