A Survey of Properties of Third Order Pell Diagonal Functions

Abstract

This paper is concerned with the simpler properties of some third order sequences of polynomials. These sequences are {r _n (x)}, {s _n (x)} and {t _n (x)}, defined thus:

$$\left. {\begin{array}{*{20}{c}} {{r_o}\left( x \right) = 0,{r_1}\left( x \right) = 1,{r_2}\left( x \right) = 2x} \\ {{r_{n + 1}}\left( x \right) = 2x{r_n}\left( x \right) + {r_{n - 2}}\left( x \right)\left( {n \geqslant 2} \right)} \end{array}} \right\},$$

((1.1))

$$\left. {\begin{array}{*{20}{c}} {{s_o}\left( x \right) = 0,{s_1}\left( x \right) = 2,{s_2}\left( x \right) = 2x} \\ {{s_{n + 1}}\left( x \right) = 2x{s_n}\left( x \right) + {s_{n - 2}}\left( x \right)\left( {n \geqslant 2} \right)} \end{array}} \right\},$$

((1.2))

$$\left. {\begin{array}{*{20}{c}} {{t_o}\left( x \right) = 3,{t_1}\left( x \right) = 2x,{t_2}\left( x \right) = 4{x^2}} \\ {{t_{n + 1}}\left( x \right) = 2x{t_n}\left( x \right) + {t_{n - 2}}\left( x \right)\left( {n \geqslant 2} \right)} \end{array}} \right\}.$$

((1.3))