On Some Results Involving Vietoris' Polynomials and Their Derivatives

Abstract

In the current study, by using a different representation of Vietoris' polynomials,  we investigate some important properties of the derivative sequence of these polynomials. We first establish some identities related to a special number sequence that we encounter as coefficients in the derivative formula. We derive fundamental recurrence relations and finite sums of these numbers. Moreover, we obtain several new recurrence-like formulas, Cassini and d'Ocagne identities related to $n$-th derivative of Vietoris' polynomials by using some combinatorial operations and inductive method.