Recurrences

A recurrence is a relationship between the terms of a sequence that allows us, recursively, to determine all the terms of the sequence, i.e. an equation of the form: \(\displaystyle \begin{aligned} \left\{\begin{array}{l}F(n, x_n, x_{n+1}, \cdots, x_{n+p})= 0,\text{ for every }n\in\mathbb{N} \\ x_0 = a_0, x_1 = a_1, \cdots, x_{p-1} = a_{p-1}\end{array},\right.\qquad \qquad {{(\ast)}} \end{aligned}\) where \(p\in \mathbb {N}^{*}, F: \mathbb {N} \times \mathbb {R}^{p+1}\to \mathbb {R}\) is a known function, and \(a_0, a_1, \cdots , a_{p-1}\in \mathbb {R}\) .