# Definition:Boubaker Polynomials

## Definition

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

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

## 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:

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

## Also see

• Results about Boubaker Polynomials can be found here.

## Source of Name

This entry was named for Boubaker Boubaker.