On the generating function of discrete Chebyshev polynomials

We give a closed form for the generating function of the discrete Chebyshev polynomials. The closed form consists of 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 the closed form implies combinatorial identities that appear quite challenging to prove directly.