Boubaker's Theorem

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 $F$.

Finally, consider the following properties:
 * $(1): \quad \displaystyle \sum_{k=1}^N{p_{n}(0)}=-2N $
 * $(2): \quad \displaystyle \sum_{k=1}^N {p_{n}(\alpha_i)}=0 $
 * $(3): \quad \displaystyle {\sum_{k=1}^N \frac {dp_{n}(x)}{dx}}_{x=0}=0 $
 * $(4): \quad \displaystyle {\sum_{k=1}^N \frac {dp_{n}(x)}{dx}_{x=\alpha_i}=\frac {8}{3}N(N^2-1)} $

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