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

0,1

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

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.

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

Example

G(2,1) = 0.9890559953279725553953956515...

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 *)

Prog

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

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.

Sequence in context: A175572 A263984 A021095 * A163930 A157371 A343060

Adjacent sequences: A090995 A090996 A090997 * A090999 A091000 A091001

Keyword

cons,nonn

Author

Benoit Cloitre, Feb 29 2004

Status

approved