Al-Qadisiyah Journal of Pure Science Vol.(26) Issue (1) (2021) pp. Math. 39–54 ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- a Department of Computer Science, University of Technology, Baghdad, Iraq, E-Mail: 110131@uotechnology.edu.iq b Department of Applied Sciences, University of Technology, Baghdad, Iraq, E-Mail: alrawy1978@yahoo.com, 100031@uotechnology.edu.iq c Department of Applied Sciences, University of Technology, Baghdad, Iraq, E-Mail: rasheed.mohammed40@yahoo.com , 10606@uotechnology.edu.iq http://qu.edu.iq/journalsc/index.php/JOPS On Some Properties of Pell Polynomials 1. Introduction Orthogonal functions and polynomial series have attention in dealing with dynamic systems' various problems, theory of elasticity, automation, and remote control [1-9]. Special class of orthogonal functions are wavelets functions, for more details, see [10- 12]. The techniques' opinion is that it reduces the dynamic system problem to solving a system of algebraic equations, which simplifies the original problem. Some approaches are based on reducing the underlying differential equation into a system of algebraic Authors Names a. Semaa Hassan Aziz b. Suha SHIHAB, c. Mohammed RASHEED Article History Received on: 4/11/2020 Revised on: 21/11/2020 Accepted on: 1/12/2020 Keywords: Pell polynomials, Expansion coefficients, Product of two polynomials, Exact formula, Power basis ABSTRACT This work starts by reviewing the Pell polynomials, its definition and some basic properties. Afterward, some new properties of such polynomials are investigated. A novel generalization analytical formula is constructed explicitly the first derivative of Pell polynomials in terms of Pell polynomials themselves. Another explicit formula is concerned with the connection between the Pell polynomials expansion coefficients; this motivates our 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 1,,  ,⋯,  are expressed in terms of Pell polynomials and an interesting formula is presented with some detail in the proof. An important general formulation for the product of two Pell polynomials is also included in this article. Explicit computations obtain all the representations in this work. Finally, two examples concern boundary value problems and singular initial value problems are included for applications of the proposed interesting properties of Pell polynomials DOI: https://doi.org/10.29350/ jops.2020.26. 1.1237