The Harmonic Logarithms and the Binomial Formula

The algebra I of polynomials in a single variable x provides a simple setting in which to do the "polynomial" calculus. Besides its being an algebra, one of the nicest features of I is that it is closed under both differentiation and antidifferentiation. That is to say, the derivative of a polynomial is another polynomial, and the antiderivative of a polynomial is another polynomial (provided we ignore the arbitrary constant). Furthermore, within the algebra 9, we have the well-known binomial formula