Abstract
The properties of the high-field polynomialsL n (u) for the one-dimensional spin 1/2 Ising model are investigated. [The polynomialsL n (u) are essentially lattice gas analogues of the Mayer cluster integralsb n (T) for a continuum gas.] It is shown thatu −1 L n (u) can be expressed in terms of a shifted Jacobi polynomial of degreen−1. From this result it follows thatu −1 L n (u); n=1, 2,... is a set of orthogonal polynomials in the interval (0, 1) with a weight functionω(u)=u, andu −1 L n (u) hasn−1 simple zerosu n (v); v=1, 2,...,n−1 which all lie in the interval 0<u<1. Next the detailed behavior ofL n (u) asn→∞ is studied. In particular, various asymptotic expansions forL n (u) are derived which areuniformly valid in the intervalsu<0, 0<u<1, andu>1. These expansions are then used to analyze the asymptotic properties of the zeros {u n (v); v=1, 2,...,n−1}. It is found that
asn→∞v fixed, wherej k,v denotes thevth zero of the Bessel functionJ k(z)
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Joyce, G.S. Asymptotic behavior of Mayer cluster sums for the one-dimensional Ising model. J Stat Phys 58, 443–465 (1990). https://doi.org/10.1007/BF01112755
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DOI: https://doi.org/10.1007/BF01112755