Pell and Pell–Lucas Polynomials

Abstract

Pell numbers and Pell–Lucas numbers are specific values of Pell polynomials p _n (x) and Pell–Lucas polynomials q _n (x), respectively. Both families were studied extensively in 1985 by A.F. Horadam of the University of New England, Armidale, Australia, and Bro. J.M. Mahon of the Catholic College of Education, Sydney, Australia [108]. Both families are often defined recursively:

$$\displaystyle\begin{array}{rcl} \begin{array}{llllllllll} p_{0}(x) & =&0,p_{1}(x) = 1 &&&&&q_{0}(x) & =&2,q_{1}(x) = 2x \\ p_{n}(x)& =&2xp_{n-1}(x) + p_{n-2}(x);&&&&&q_{n}(x)& =&2xq_{n-1}(x) + q_{n-2}(x),\end{array} & & {}\\ \end{array}$$

where n ≥ 3. The first ten Pell and Pell–Lucas polynomials are given in Table 14.1.