On (self-) reciprocal Appell polynomials: Symmetry and Faulhaber-type polynomials

The main purpose of this paper is to study generalized (self-) reciprocal Appell polynomials, which play a certain role in connection with Faulhaber-type polynomials. More precisely, we show for any Appell sequence when satisfying a reflection relation that the Appell polynomials can be described by Faulhaber-type polynomials, which arise from a quadratic variable substitution. Furthermore, the coefficients of the latter polynomials are given by values of derivatives of generalized reciprocal Appell polynomials. Subsequently, we show some applications to the Bernoulli and Euler polynomials. In the context of power sums the results transfer to the classical Faulhaber polynomials.