Definition:Boubaker Polynomials

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

They can be seen to be a special case of the Chebyshev polynomials.

In fact,like Lucas polynomials, Dickson polynomials and Fibonacci polynomials , Boubaker polynomials are related to Chebyshev polynomials  and Un by:
 * $$B_n(2x)= \frac{4x}{n}\frac{d}{dx}T_n(x)-2T_n(x) \, $$
 * $$B_n(2x)= -2T_n(x)+4xU_{n-1}(x) \, $$
 * The Boubaker polynomials Bn  are also linked  to the  Dickson polynomials  by the relations :
 * $$B_{n+1}(x)B_{n+j}(x)-B_{n+j+1}(x)B_{n}(x)=(3x^2+4)D_{n+1}(x,\frac{1}{4}) \, $$
 * $$B_n \left(x \right)= D_n(2x,\frac{1}{4})+4D_{n-1}(2x,\frac{1}{4}) \, $$


 * A . Luzon et al. demonstrated, thanks to Riordan matrices analysis, the link between  the Boubaker polynomials  Bn  and the Fermat polynomials Fn:
 * $$B_{n}(x)=\frac{1}{({\sqrt {2})}^n}F_n(\frac{2\sqrt {2}x}{3})+\frac{3}{{(\sqrt {2})}^{n-2}}F_{n-2}(\frac{2\sqrt {2}x}{3}),

n=0,1,2,..., $$

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:


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