On Some Properties of Pell Polynomials

Abstract

This work starts by reviewing the Pell polynomials; its definition and some basic properties. Afterwards, some new properties of such polynomials are investigated. A novel generalization analytical formula is provided defining explicitly the Pell polynomials derivatives of order n in terms of  Pell polynomials themselves. Other explicit formula is concerned with the connection between the Pell polynomials expansion coefficients, this motivates are interest in such polynomials. These formulas are utilized to derive some mainly relationship related with power basis coefficients and Pell polynomials. With the Pell polynomials expansion technique, the powers  are expressed in terms of Pell polynomials and an interesting formula is presented with some detail in the proof. An important general formulation for product of two Pell polynomials is also included in this article. All the representations in this work are obtained by explicit computations. Finally, two examples concern boundary value problem and singular initial value problem are included for applications of the proposed interesting properties of Pell polynomials.