Generalized Exponential and Trigonometric Functions

Abstract

One of the most classical papers written on generalized Fibonacci numbers is that of A. F. Horadam, [8]. In that paper, one will find several of the following properties needed in order to obtain the results found in this article. We start with a special case of the basic definition found in [8]. That is, we let

$$ {G_0} = 0,\;{G_1} = 1,\;{G_{{n + 2}}} = P{G_{{n + 1}}} + Q{G_n},\;n \geqslant 0, $$

(1.1)

where P and Q are integers. The Binet formula associated with (1) is

$$ {G_n} = \frac{{{\alpha^n} - {\beta^n}}}{{\alpha - \beta }} $$

(1.2)

where α is the positive root of x² − Px − Q = 0 and β is the negative root. The generalized Lucas sequence associated with {G _n } is given by the formula

$$ {H_n} = {\alpha^n} + {\beta^n}. $$

(1.3)

.