Some Results on a Two Variables Pell Polynomials

Abstract

In this study, Pell polynomials in two variables, and their properties are investigated. Some formulas for two variables Pell polynomials are derived by matrices. By defining special formula for Pell polynomials in one variable, new important properties of Pell polynomials in two variables can be enabled to derive. A new exact formula expressing the partial derivatives of Pell polynomials explicitly of any degree in terms of Pell polynomials themselves is proved. A novel explicit formula, which constructs the two explicit formulas, which construct the two-dimension Pell polynomials expansion coefficients of a first partial derivative of a differentiable function in terms of their original expansion coefficients, is also included in the present article. The main advantage of the presented formulas is that the new properties of Pell polynomials in two variables greatly simplify the original problems and the result will lead to easy calculation of the coefficients of expansion. A direct spectral method along with the presented two variables Pell polynomials is proposed to solve the partial differential equation. Illustrated examples are included to demonstrate the validity of the technique.