New Aspects of Morgan-Voyce Polynomials

Abstract

What formal connection, if any, is there between the equation

$$ {\mu ^{2n}} - {L_n}{\mu ^n} + {( - 1)^n} = 0 $$

(1.1)

(where L _n is the n ^th Lucas number) and the Morgan-Voyce polynomials B _n(x) and b _n(x), to be defined in (2.1)–(2.6)? From the Binet forms [7] for L _n and F _n (the n ^th Fibonacci number), we readily have

$$ _{\beta = {\rm{ }} = \frac{{{L_n} - \sqrt {5{F_n}} }}{2}}^{\alpha n = \frac{{{L_n} - \sqrt {5{F_n}} }}{2}}\left. {} \right\} $$

(1.2)

where

$$ \alpha = \frac{{1 + \sqrt 5 }}{2},{\rm{ }}\beta = \frac{{1 - \sqrt 5 }}{2} $$

(1.3)

are the roots of the characteristic quadratic equation t ² - t - 1 = 0 for both F _n and L _n. Thus, αⁿ and βⁿ are the roots of (1.1) or, alternatively, of

$$ {v^{2n}} - \sqrt {5{F_n}{v^n} + {{( - 1)}^{n + 1}} = 0} $$

(1.4)

since [7]

$$ 5F_n^2 - L_n^2 = 4{( - 1)^{n + 1}} $$

(1.5)

.