Definition:Boubaker Polynomials

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

Recursive Definition
The Boubaker polynomials  are defined as:
 * $B_n \left({x}\right) = \begin{cases}

1 & : n = 0 \\ x & : n = 1 \\ x^2+2 & : n = 2 \\ x B_{n-1} \left({x}\right) - B_{n-2} \left({x}\right) & : n > 2 \end{cases}$

Closed Form
The Boubaker polynomials  are defined in closed form as:
 * $\displaystyle B_n \left({x}\right) = \sum_{p \mathop = 0}^{\left\lfloor{n/2}\right\rfloor} \frac {n - 4 p} {n - p} \binom {n - p} p \left({-1}\right)^p x^{n - 2 p}$

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


 * $\left({x^2 - 1}\right) \left({3 n x^2 + n - 2}\right) \dfrac {\mathrm d^2 y} {\mathrm d x^2} + 3 x \left({n x^2 + 3 n - 2}\right) \dfrac {\mathrm d y}{\mathrm d x} - n \left({3 n^2 x^2 + n^2 - 6 n + 8}\right) y = 0$

Also see

 * Equivalence of Definitions of Boubaker Polynomials