m = 7/34169*10^(-6)*mp;
mp = 2.435*10^(18);
k = 8*10^(-41);
H = 1; (* was missing *)
NDSolve[{u''[n] + (-2/n^2 + k^2 + m^2/H^2/n^2)*u[n] == 0,
Normal[Series[u[n], {n, Infinity, 0}]] == Exp[(2*k)^(1/2)]}, u, {n,
8*10^(38), 1.13*10^(42)}]
NDSolve::overdet: There are fewer dependent variables, {u[n]}, than equations, so the system is overdetermined. >>
is this meant to be a differential equation (why then there are conditions {n, Infinity, 0} outside the range {n, 8 10^38, 1.13 10^42) or is this meant to be a recurrence equation (why then are the derivatives)?