Pell and Pell-Lucas polynomials. (English) Zbl 0557.10011

Horadam, Alwyn F.; Mahon, Brother J. M.

Fibonacci Q. 23, 7-20 (1985).

Pell polynomials \(P_ n(x)\) and Pell-Lucas polynomials \(Q_ n(x)\) are defined by the recurrence relations \(P_{n+2}(x)=2\times P_{n+1}(x)+P_ n(x)\), \(P_ 0(x)=0\), \(P_ 1(x)=1\) and \(Q_{n+2}(x)=2\times Q_{n+1}(x)+Q_ n(x)\), \(Q_ 0(x)=2\), \(Q_ 1(x)=2\), respectively. Basic properties of these polynomials are established by a variety of means, one of the most fruitful being the use of matrices. The relationships of \(P_ n(x)\) and \(Q_ n(x)\) to some classical polynomials, such as the Gegenbauer polynomial and the Chebyshev polynomials, are briefly investigated.

MSC:

11B39	Fibonacci and Lucas numbers and polynomials and generalizations
11B37	Recurrences

Keywords:

Pell polynomials; Pell-Lucas polynomials; Basic properties; matrices; Gegenbauer polynomial; Chebyshev polynomials