The Abel-Type Polynomial Identities

Abstract

The Abel identity is $(x+y)^{n}=\sum\limits_{i=0}^{n}{n\choose i}x (x-iz)^{i-1}(y+iz)^{n-i}$, where $x,y$ and $z$ are real numbers. In this paper we deduce several polynomials expansions, referred to as Abel-type identities, by using Foata's method, and also show some of their applications.