Boubaker's Theorem

Theorem

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

Let $\left({D, +, \circ}\right)$ be an integral subdomain 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 $D$.

Finally, consider the following properties:

\(\text {(1)}: \quad\)

\(\displaystyle \sum_{k \mathop = 1}^N {p_n \left({0}\right)}\)

\(=\)

\(\displaystyle -2N\)

\(\text {(2)}: \quad\)

\(\displaystyle \sum_{k \mathop = 1}^N {p_n \left({\alpha_k}\right)}\)

\(=\)

\(\displaystyle 0\)

\(\text {(3)}: \quad\)

\(\displaystyle \left.{\sum_{k \mathop = 1}^N \frac {\mathrm d p_x \left({x}\right)} {\mathrm d x} }\right\vert_{x \mathop = 0}\)

\(=\)

\(\displaystyle 0\)

\(\text {(4)}: \quad\)

\(\displaystyle \left.{\sum_{k \mathop = 1}^N \frac {\mathrm d {p_n}^2 \left({x}\right)} {\mathrm d x^2} }\right\vert_{x \mathop = 0}\)

\(=\)

\(\displaystyle \frac 8 3 N \left({N^2 - 1}\right)\)

where, for a given positive integer $n$, $p_n \in D \left[{X}\right]$ is a non-null polynomial such that $p_n$ has $N$ roots $\alpha_k$ in $F$.

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

Proof

Proof of validity

We first prove that the Boubaker Polynomials sub-sequence $ B_{4n}(x)$, defined in $D \left[{X}\right]$ verifies properties $(1)$, $(2)$, $(3)$ and $(4)$.

Let:

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

$\left({D, +, \circ}\right)$ be an integral subdomain of $R$ whose zero is $0_D$ and whose unity is $1_D$

$X \in R$ be transcendental over $D$.

Property $(1)$

We have the closed form of the the Boubaker Polynomials:

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

which gives:

$B_{4 n} \left({0}\right) = \dfrac {n - 4 n / 2} {n - n / 2} \dbinom {n - n / 2} {n / 2} = -2$

and finally:

$(1): \quad \displaystyle \sum_{k \mathop = 1}^N B_{4 n} \left({0}\right) = \sum_{k \mathop = 1}^N -2 = - 2 N$

Property $(2)$

We have, for given integer $n$, $ B_{4n} \in D \left[{X}\right]$ is a non-null polynomial with $N$ roots $\alpha_k$ in $F$.

Since:

$B_{4 n} \left({\alpha_k}\right) = 0$

then the equality:

$(2): \quad \displaystyle \sum_{k \mathop = 1}^N B_{4 n} \left({\alpha_k}\right) = 0$

holds.

Property $(3)$

According to the closed form of the the Boubaker Polynomials:

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

We have:

$\displaystyle \frac {\mathrm d B_{4 n} \left({x}\right)} {\mathrm d x} = \sum_{p \mathop = 0}^{\left\lfloor{n / 2}\right\rfloor - 1} \frac {n - 4 p} {n - p} \binom {n - p} p \left({-1}\right)^p \left({n - 2 p}\right) x^{n - 2 p - 1}$

The minimal power in this expansion is obtained for $p = {\left\lfloor{n / 2}\right\rfloor - 1}$, hence:

$\dfrac {\mathrm d B_{4 n} } {\mathrm d x} \left({0}\right) = 0$

and the equality:

$(3): \quad \displaystyle \left.{\sum_{k \mathop = 1}^N \frac {\mathrm d B_{4 n} \left({x}\right)} {\mathrm d x} } \right\vert_{x \mathop = 0} = 0$

holds.

Property $(4)$

Starting from the closed form of the the Boubaker Polynomials:

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

we have consequently:

$\displaystyle \frac {\mathrm d^2 B_{4n} \left({x}\right)} {\mathrm d x^2} = \sum_{p \mathop = 0}^{\left\lfloor{n / 2}\right\rfloor - 2} \frac {n - 4 p} {n - p} \binom {n - p} p \left({-1}\right)^p \left({n - 2 p}\right) \left({n - 2 p - 1}\right) x^{n - 2 p - 2}$

The minimal power in this expansion is obtained for $p = \left\lfloor{n / 2}\right\rfloor - 2$, hence:

$\displaystyle \frac {\mathrm d^2 B_{4 n} } {\mathrm d x^2} \left({0}\right) = \left({-1}\right)^p \left({n - 2 p}\right) \left({n - 2 p - 1}\right) 0$

and the equality:

$(4): \quad \displaystyle \left.{\sum_{k \mathop = 1}^N \frac {\mathrm d^2 B_{4 n} \left({x}\right)} {\mathrm d x^2}}\right\vert_{x \mathop = 0} = \frac 8 3 N \left({N^2 - 1}\right)$

holds.

$\blacksquare$

Proof of Uniqueness

Let:

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

$\left({D, +, \circ}\right)$ be an integral subdomain of $R$ whose zero is $0_D$ and whose unity is $1_D$

$X \in R$ be transcendental over $D$.

It has been demonstrated that the Boubaker Polynomials sub-sequence $B_{4 n} \left({x}\right)$, defined in $D \left[{X}\right]$ as:

$\displaystyle B_{4 n} \left({x}\right) = 4 \sum_{p \mathop = 0}^{2 n} \frac {n - p} {4 n - p} \binom {4 n - p} p \left({-1}\right)^p x^{2 \left({2n - p}\right)}$

satisfies the properties:

\(\text {(1)}: \quad\)

\(\displaystyle \sum_{k \mathop = 1}^N {p_n \left({0}\right)}\)

\(=\)

\(\displaystyle -2N\)

\(\text {(2)}: \quad\)

\(\displaystyle \sum_{k \mathop = 1}^N {p_n \left({\alpha_k}\right)}\)

\(=\)

\(\displaystyle 0\)

\(\text {(3)}: \quad\)

\(\displaystyle \left.{\sum_{k \mathop = 1}^N \frac {\mathrm d p_x \left({x}\right)} {\mathrm d x} }\right\vert_{x \mathop = 0}\)

\(=\)

\(\displaystyle 0\)

\(\text {(4)}: \quad\)

\(\displaystyle \left.{\sum_{k \mathop = 1}^N \frac {\mathrm d {p_n}^2 \left({x}\right)} {\mathrm d x^2} }\right\vert_{x \mathop = 0}\)

\(=\)

\(\displaystyle \frac 8 3 N \left({N^2 - 1}\right)\)

with $\left. {\alpha_k}\right\vert_{k \mathop = 1 \,.\,.\, N}$ roots of $B_{4 n}$.

Suppose there exists another $4 n$-indexed polynomial $q_{4 n} \left({x}\right)$, with $N$ roots $\left.{\beta_k}\right\vert_{k \mathop = 1 \,.\,.\, N}$ in $F$ and which also satisfies simultaneously properties $(1)$ to $(4)$.

Let:

$\displaystyle B_{4 n} \left({x}\right) = \sum_{p \mathop = 0}^{2 n} a_{4 n, p} x^{2 \left({2 n - p}\right)}$

and:

$\displaystyle q_{4 n} \left({x}\right) = \sum_{p \mathop = 0}^{2 n} b_{4 n, p} x^{2 \left({2 n - p}\right)}$

and:

$\displaystyle \mathrm d_{4 n, p} = a_{4 n, p} - b_{4 n, p}$ for $p = 0 \,.\,.\, 2 n$

then, simultaneous expressions of conditions $(1)$ and $(3)$ give:

$ \quad \displaystyle \sum_{k \mathop = 1}^N \mathrm d_{4 n, 2 n} = 0$

$ \quad \displaystyle \sum_{k \mathop = 1}^N \mathrm d_{4 n, 2 n - 2} = 0$

It has also been demonstrated that $ B_{4 n}$ has exactly $4 n - 2$ real roots inside the domain $\left[{-2 \,.\,.\, 2}\right]$.

So application of conditions $(3)$ and $(4)$ give $4n-2$ linear equation with variables $\left.{ d_{4n,p}}\right|_{p \mathop = 0 \,.\,.\, 2n-3}$.

Finally, since $B_{4 n}$ contains $2 n$ monomial terms (see definition), we obtain a Cramer system in variables $\left.{ d_{4 n, p}}\right\vert_{p \mathop = 0 \,.\,.\, 2 n}$, with evident solution:

$\left.{\mathrm d_{4 n, p}}\right\vert_{p \mathop = 0 \,.\,.\, 2 n} = 0 $

and consequently:

$\left.{a_{4 n, p} }\right\vert_{p \mathop = 0 \,.\,.\, 2 n} = \left.{b_{4 n, p} }\right\vert_{p \mathop = 0 \,.\,.\, 2 n}$

which means:

$q_{4 n} \left({x}\right) = B_{4 n} \left({x}\right)$

$\blacksquare$

Source of Name

This entry was named for Boubaker Boubaker.