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We study a linear spin chain which was originally introduced by Shi et al.[Phys. Rev. A 71 (2005), 032309, 5 pages], for which the coupling strengthcontains a parameter $\alpha$ and depends on the parity of the chain site.Extending the model by a second parameter $\beta$, it is shown that the singlefermion eigenstates of the Hamiltonian can be computed in explicit form. Thecomponents of these eigenvectors turn out to be Hahn polynomials withparameters $(\alpha,\beta)$ and $(\alpha+1,\beta-1)$. The construction of theeigenvectors relies on two new difference equations for Hahn polynomials. Theexplicit knowledge of the eigenstates leads to a closed form expression for thecorrelation function of the spin chain. We also discuss some aspects of a$q$-extension of this model.

An Exactly Solvable Spin Chain Related to Hahn Polynomials