0,5

The rationals A(n) defined below appear in the expansion of the one-loop effective potential V1(y) for the thermal phi^4 model. See the Dolan-Jackiw, Kapusta and Quir/'os references. The expansion variable is y:=(m^2(phi))/(2*pi*k*T)^2 with Boltzmann's constant k, the (absolute) temperature T and m^2(phi):= m^2 + (lambda/2) phi^2 if the coupling constant is lambda/4! and the mass is m.

The relevant expansion of part of the thermal one-loop effective potential is ((pi^2)*((k*T)^4)/2)*sum(A(n)*Zeta(2*n+1)*(-1)^(n+1)*y^(n+2),n=1..infty) with Riemann's Zeta function. The expansion parameter y is given above. See the W. Lang link for more details.

L. Dolan and R. Jackiw, Symmetry behavior at finite temperature, Phys.Rev. D9,12 (1974) 3320-41.

J. I. Kapusta, Finite-temperature field theory, Cambridge University Press, 1989.

M. Quir/'os, Field theory at finite temperature and phase transitions, Helv. Phys. Acta 67 (1994) 451-583.

W. Lang: Rationals A(n) and more.

a(n)= numerator(A(n)) with A(n):= Catalan(n)/((n+2)*2^(2*n-1)) where Catalan(n):=A000108(n)=binomial(2*n, n)/(n+1).

a(n)= numerator(8*(2*n-1)!!/((2*(n+2))!!)) with the double factorials (2*n-1)!!:=A001147(n) (with (-1)!!:=1) and (2*n)!!:=A000165(n).

Rationals A(n):=A099398(n)/A099399(n), n>=0: 1/1, 1/6, 1/16, 1/32, 7/384, ...

The denominators are given in A099399.

Adjacent sequences: A099395 A099396 A099397 this_sequence A099399 A099400 A099401

Sequence in context: A002969 A104727 A116419 this_sequence A038269 A003723 A054471

nonn,frac,easy

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Nov 10 2004