We give a closed form for the generating function of the discrete Chebyshev polynomials. It is the MacWilliams transform of Jacobi polynomials together with a binomial multiplicative factor. It turns out that the desired closed form is a solution to a special case of the Heun differential equation, and that it implies combinatorial identities that appear quite challenging to prove directly.
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 448, 2016, pp. 124–134.
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Gogin, N., Hirvensalo, M. On the Generating Function of Discrete Chebyshev Polynomials. J Math Sci 224, 250–257 (2017). https://doi.org/10.1007/s10958-017-3410-8
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DOI: https://doi.org/10.1007/s10958-017-3410-8