Journal of Statistical Physics, Vol. 82, Nos. 1/2, 1996. p. 343-365.
We present a differential formulation of the recursion formula of the hierarchical model which provides a recursive method of calculation for the high-temperature expansion. We calculate the first 30 coefficients of the high-temperature expansion of the magnetic susceptibility of the Ising hierarchical model with 12 significant digits. We study the departure from the approximation which consists in identifying the coefficients with the values they would take if a [0, 1] Pade approximant were exact. We show that, when the order in the high-temperature expansion increases, the departure from this approximation grows more slowly than for nearest neighbor models. As a consequence, the value of the critical exponent y estimated using Pade approximants converges very slowly and the estimations using 30 coefficients have errors larger than 0.05. A (presumably much) larger number of coefficients is necessary to obtain the critical exponents with a precision comparable to the precision obtained for nearest neighbor models with fewer coefficients. We also discuss the possibility of constructing models where a [0, 1] Pade approximant would be exact.