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In this paper we follow the general approach, proposed earlier by the firstauthor, which is derived from the invariant theory field and provides a way ofobtaining of the polynomial identities for any arbitrary polynomial family. Weintroduce the notion of Chebyshev derivations of the first and second kinds,which is based on the polynomial algebra, and corresponding specificdifferential operators. We derive the elements of their kernels and prove thatany element of the kernel of the derivations defines a polynomial identitysatisfied by the Chebyshev polynomials of the first and second kinds. Combiningelementary methods and combinatorial techniques, we obtain several newpolynomial identities involving the Chebyshev polynomials of the both kinds anda special case of the Jacobi polynomials. Using the properties of thegeneralised hypergeometric function, we specify the Chebyshev polynomials ofthe first and second kinds via the generalised hypergeometric function and, asa consequence, derive the corresponding identities involving the generalisedhypergeometric function and the Chebyshev polynomials of the first and secondkinds.

Derivations and identities for Chebyshev polynomials of the first and second kinds