Infinite-Spin Ising Model in One Dimension

The partition function z, the pair correlation function ρ, and the zero-field susceptibility χ for the one-dimensional Ising model with infinite spin, are expressed in terms of the eigenvalues and eigenfunctions of an integral equation. The eigenfunctions of the integral equation are shown to be the oblate spheroidal wavefunctions, and, from known asymptotic expansions, high- and low-temperature expansions are given for z, ρ, and χ. It is shown that the low-temperature behavior of z, ρ, and χ differs qualitatively from the corresponding behavior for all finite spin.