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    Structure of Certain Chebyshev-type Polynomials in Onsager's Algebra Representation

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    In this report, we present a systematic account of mathematical structures of certain special polynomials arisen from the energy study of the superintegrable NN-state chiral Potts model with a finite number of sizes. The polynomials of low-lying sectors are represented in two different forms, one of which is directly related to the energy description of superintegrable chiral Potts \ZZ_N-spin chain via the representation theory of Onsager's algebra. Both two types of polynomials satisfy some (N+1)(N+1)-term recurrence relations, and NNth order differential equations; polynomials of one kind reveal certain Chebyshev-like properties. Here we provide a rigorous mathematical argument for cases N=2,3N=2, 3, and further raise some mathematical conjectures on those special polynomials for a general NN.Comment: 18 pages, Latex ; Typos corrected, Small changes for clearer presentatio
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