Definition:Boubaker Polynomials

From ProofWiki
Jump to: navigation, search

Definition

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

\(\displaystyle B_0 \left({x}\right)\) \(=\) \(\displaystyle 1\) $\quad$ $\quad$
\(\displaystyle B_1 \left({x}\right)\) \(=\) \(\displaystyle x\) $\quad$ $\quad$
\(\displaystyle B_2 \left({x}\right)\) \(=\) \(\displaystyle x^2 + 2\) $\quad$ $\quad$
\(\displaystyle B_3 \left({x}\right)\) \(=\) \(\displaystyle x^3 + x\) $\quad$ $\quad$
\(\displaystyle B_4 \left({x}\right)\) \(=\) \(\displaystyle x^4 - 2\) $\quad$ $\quad$
\(\displaystyle B_5 \left({x}\right)\) \(=\) \(\displaystyle x^5 - x^3 - 3x\) $\quad$ $\quad$
\(\displaystyle B_6 \left({x}\right)\) \(=\) \(\displaystyle x^6 - 2x^4 - 3x^2 + 2\) $\quad$ $\quad$
\(\displaystyle B_7 \left({x}\right)\) \(=\) \(\displaystyle x^7 - 3x^5 - 2x^3 + 5x\) $\quad$ $\quad$
\(\displaystyle B_8 \left({x}\right)\) \(=\) \(\displaystyle x^8 - 4x^6 + 8x^2 - 2\) $\quad$ $\quad$
\(\displaystyle B_9 \left({x}\right)\) \(=\) \(\displaystyle x^9 - 5x^7 + 3x^5 + 10x^3 - 7x\) $\quad$ $\quad$
\(\displaystyle \) \(\vdots\) \(\displaystyle \) $\quad$ $\quad$


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:

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


From Differential Equation

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

$\left({x^2 - 1}\right) \left({3 n x^2 + n - 2}\right) \dfrac {\mathrm d^2 y} {\mathrm d x^2} + 3 x \left({n x^2 + 3 n - 2}\right) \dfrac {\mathrm d y}{\mathrm d x} - n \left({3 n^2 x^2 + n^2 - 6 n + 8}\right) y = 0$


Also see


Source of Name

This entry was named for Boubaker Boubaker.