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 \\ B_{n-1} \left({x}\right) - xB_{n-2} \left({x}\right) & : n > 2 \end{cases}$

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

Boubaker Theorem (to demonstrate)
Boubaker's Theorem


 * Let $\left({R, +, \circ}\right)$ be a commutative ring.


 * Let $\left({D, +, \circ}\right)$ be an integral domain such that $D$ is a subring of $R$ whose zero is $0_D$ and whose unity is $1_D$.


 * Let $X \in R$ be transcendental over $D$.
 * Let $D \left[{X}\right]$ be the ring of polynomial forms in $X$ over $D$.
 * For a given integer $n$, if $p_n\in F[x]$ be non-null, and if $p_n$ has $N$ roots $\alpha_i$ in $F$.

If, finally, we consider the following properties:
 * $$ \sum_{n=1}^N B_{4n} \left( \frac{x}{a}\alpha_n \right) \left.{\!\!\frac{}{}}\right|_{x=0} =-2N$$
 * $$ \sum_{n=1}^N B_{4n} \left( \frac{x}{a}\alpha_n \right) \left.{\!\!\frac{}{}}\right|_{x=\pm a} =0$$
 * $$ \sum_{n=1}^N \frac{d}{dx} \left(B_{4n} \left( \frac{x}{a}\alpha_n \right)\right) \left.{\!\!\frac{}{}}\right|_{x=0} =0$$
 * $$ \sum_{n=1}^N \frac{d}{dx} \left(B_{4n} \left( \frac{x}{a}\alpha_n \right)\right) \left.{\!\!\frac{}{}}\right|_{x=\pm a} =\sum_{q=1}^{N} ( \displaystyle\frac {4\alpha_q (2-\alpha_q^2)\times \sum_{j=1}^{q} B_{4j}^{2}(\alpha_q)}{B_{4(q+1)}(\alpha_q)}+4\alpha_q^3)=\sum_{q=1}^{N}H_q$$
 * $$ \sum_{n=1}^N \frac{d^2}{d^2 x} \left(B_{4n} \left( \frac{x}{a}\alpha_n \right)\right) \left.{\!\!\frac{}{}}\right|_{x=0} =8N(N^2-1)$$
 * $$ \sum_{n=1}^N \frac{d^2}{d^2 x} \left(B_{4n} \left( \frac{x}{a}\alpha_n \right)\right) \left.{\!\!\frac{}{}}\right|_{x=\pm a} =\sum_{q=1}^{N} \displaystyle\frac {3\alpha_q (4q \alpha_q^2+12\alpha_q-2)\times H_q-8q(24q^{2}\alpha_q^2+8\alpha_q^2-3q+4)}{(\alpha_q^{2}-1)(12q\alpha_q^{2}+4q-2)}$$

Then the Boubaker Polynomials sub-sequence $B_{4n}(x)$ is the unique polynomial sequence of $D \left[{X}\right]$ which verifies simultaneously the four properties (1-4).

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


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