Definition:Boubaker Polynomials

Definition
The Boubaker polynomials are the components of the following sequence of polynomials:

Recursive Definition
The Boubaker polynomials  are defined as:
 * $\map {B_n} x = \begin{cases}

1 & : n = 0 \\ x & : n = 1 \\ x^2+2 & : n = 2 \\ x \map {B_{n - 1} } x - \map {B_{n - 2} } x & : n > 2 \end{cases}$

Closed Form
The Boubaker polynomials  are defined in closed form as:
 * $\ds \map {B_n} x = \sum_{p \mathop = 0}^{\floor {n / 2} } \frac {n - 4 p} {n - p} \binom {n - p} p \paren {-1}^p x^{n - 2 p}$

From Differential Equation
The Boubaker polynomials  are defined as solutions to the differential equation:


 * $\paren {x^2 - 1} \paren {3 n x^2 + n - 2} \dfrac {\rd^2 y} {\rd x^2} + 3 x \paren {n x^2 + 3 n - 2} \dfrac {\rd y} {\rd x} - n \paren {3 n^2 x^2 + n^2 - 6 n + 8} y = 0$

Also see

 * Equivalence of Definitions of Boubaker Polynomials