On Appell-Vietoris Polynomials

In this paper we introduce a polynomial sequence \((\mathcal {V}_n(x))_{n\ge 0}\) involving the Vietoris’ number sequence and satisfying the Appell property. We derive several equivalent representations of \(\mathcal {V}_n(x)\) and we deduce important relations closely related to the aforementioned Appell property. Finally, we address the problem of the zeros distribution of \(\mathcal {V}_n(x)\) and conclude that, when n is even, the polynomials \(\mathcal {V}_n(x)\) have no real zeros, whereas for odd n, the unique real zero of \(\mathcal {V}_n(x)\) lies within the interval \((-1,0)\) .