A274181 Decimal expansion of Phi(1/2, 2, 2), where Phi is the Lerch transcendent.

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

Offset: 0

Author

Johannes W. Meijer and N. H. G. Baken, Jun 17 2016, Jul 08 2016

Keywords

• cons
• nonn

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.

Examples

			0.32896210586005002361062528063872043497679389922...
		

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.

Crossrefs

Cf. A163931, A002162 (Phi(1/2, 1, 1)/2), A076788 (Phi(1/2, 2, 1)/2), A112302, A008276.

Programs

Maple

Digits := 101; c := evalf(LerchPhi(1/2, 2, 2));

Mathematica

N[HurwitzLerchPhi[1/2, 2, 2], 25] (* G. C. Greubel, Jun 19 2016 *)

PARI

Pi^2/3 - 2*log(2)^2 - 2 \\ Altug Alkan, Jul 08 2016

PARI

lerchphi(.5,2,2) \\ Charles R Greathouse IV, Jan 30 2025

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

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