Hyperdifferential and Symbolic Representations of Chebyshev Polynomials

Abstract

This work shows that Chebyshev polynomials of the first kind T_n (X) and of the second kind U_n (X) may be represented as the transforms of the monomial Xⁿ each by a hyperdifferential operator so that they may be calculated easily from a symbolic formula similar to the Lucas formula for Bernoulli polynomials. It exposes also a new approach for obtaining their generating functions by operator calculus built from the derivative \(\partial\)_x and the “multiply by x” operators instead of fastidious summations.