OFFSET
0,1
COMMENTS
The exponential integral distribution is defined by p(x, m, n, mu) = ((n+mu-1)^m * x^(mu-1) / (mu-1)!) * E(x, m, n), see A163931 and the Meijer link. The moment generating function of this probability distribution function is M(a, m, n, mu) = Sum_{k>=0}(((mu+k-1)!/((mu-1)!*k!)) * ((n+mu-1) / (n+mu+k-1))^m * a^k).
In the special case that mu=1 we get p(x, m, n, mu=1) = n^m * E(x, m, n) and M(a, m, n, mu=1) = n^m * Phi(a, m, n), with Phi the Lerch transcendent. If n=1 and mu=1 we get M(a, m, n=1, mu=1) = polylog(m, a)/a = Li_m(a)/a.
REFERENCES
William Feller, An introduction to probability theory and its applications, Vol. 1. p. 285, 1968.
LINKS
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.
Eric W. Weisstein’s World of Mathematics, Lerch transcendent.
Eric W. Weisstein’s World of Mathematics, Polylogarithm.
FORMULA
Equals Phi(1/2, 2, 2) with Phi the Lerch transcendent.
Equals Sum_{k>=0}(1/((2+k)^2*2^k)).
Equals 4 * polylog(2, 1/2) - 2.
Equals Pi^2/3 - 2*log(2)^2 - 2.
Equals Integral_{x=0..oo} x*exp(-x)/(exp(x)-1/2) dx. - Amiram Eldar, Aug 24 2020
EXAMPLE
0.32896210586005002361062528063872043497679389922...
MAPLE
Digits := 101; c := evalf(LerchPhi(1/2, 2, 2));
MATHEMATICA
N[HurwitzLerchPhi[1/2, 2, 2], 25] (* G. C. Greubel, Jun 19 2016 *)
PROG
(PARI) Pi^2/3 - 2*log(2)^2 - 2 \\ Altug Alkan, Jul 08 2016
(Python)
from mpmath import mp, lerchphi
mp.dps=102
print([int(d) for d in list(str(lerchphi(1/2, 2, 2))[2:-1])]) # Indranil Ghosh, Jul 04 2017
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Johannes W. Meijer and N. H. G. Baken, Jun 17 2016, Jul 08 2016
STATUS
approved