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:
 * $\displaystyle B_n \left({x}\right) = \sum_{p=0}^{\lfloor n/2\rfloor} \frac {n-4p} {n-p} \binom {n-p} p \left({-1}\right)^p x^{n-2p}$

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


 * $\displaystyle \left({x^2-1}\right) \left({3nx^2+n-2}\right) \frac {d^2y} {dx^2} + 3x \left({n x^2 + 3n - 2}\right) \frac {dy}{dx} - n \left({3n^2 x^2 + n^2 - 6n+8}\right) y = 0$

Also see

 * Boubaker's Theorem
 * Relation of Boubaker Polynomials to Chebyshev Polynomials
 * Relation of Boubaker Polynomials to Dickson Polynomials
 * Relation of Boubaker Polynomials to Fermat Polynomials